Systematic time expansion for the Kardar-Parisi-Zhang equation, linear statistics of the GUE at the edge and trapped fermions

We present a systematic short time expansion for the generating function of the one point height probability distribution for the KPZ equation with droplet initial condition, which goes much beyond previous studies. The expansion is checked against a numerical evaluation of the known exact Fredholm determinant expression. We also obtain the next order term for the Brownian initial condition. Although initially devised for short time, a resummation of the series allows to obtain also the \textit{long time large deviation function}, found to agree with previous works using completely different techniques. Unexpected similarities with stationary large deviations of TASEP with periodic and open boundaries are discussed. Two additional applications are given. (i) Our method is generalized to study the linear statistics of the {Airy point process}, i.e. of the GUE edge eigenvalues. We obtain the generating function of the cumulants of the empirical measure to a high order. The second cumulant is found to match the result in the bulk obtained from the Gaussian free field by Borodin and Ferrari, but we obtain systematic corrections to the Gaussian free field (higher cumulants, expansion towards the edge). This also extends a result of Basor and Widom to a much higher order. We obtain {large deviation functions} for the {Airy point process} for a variety of linear statistics test functions. (ii) We obtain results for the \textit{counting statistics of trapped fermions} at the edge of the Fermi gas in both the high and the low temperature limits.


A. Overview
Recent exact results on a number of models in the 1D Kardar-Parisi-Zhang (KPZ) universality class allows insight into non-equilibrium dynamics for a broad class of systems [4]. Such models [5][6][7][8] range from discrete particle transport models, such as the TASEP [9], to continuum models of stochastic interface growth, such as the KPZ equation [10][11][12][13]. In all of them, an analog of a "height field" can be defined, with common universal behavior at large scale defining the KPZ class. Although exact formulas valid at all time exist for some observables, such as the probability distribution function (PDF) of the (centered) height H(t) at a given space-time point for some initial conditions, extracting from them the physical information about the stochastic process may prove quite difficult.
It often amounts to study functional (i.e. Fredholm) determinants, which can be evaluated numerically [14], but from which explicit formulas are hard to extract. While the long time asymptotics for the typical fluctuations (i.e. the regime where H(t) ∼ t 1/3 ) has been obtained in several cases, leading to the celebrated Tracy-Widom distribution of random matrix theory [15], the short and intermediate time behavior, as well as the regime of large fluctuations has only been studied recently.
The short time expansion of the KPZ equation formally identifies with perturbation theory in the noise. Naive perturbation theory straightforwardly shows that typical behavior is similar to its linear version, the Edwards-Wilkinson (EW) equation, with Gaussian fluctuations H(t) ∼ t 1/4 . Study of the higher cumulants of the height shows, however, that there exists a non trivial short time large deviation regime, H(t) ∼ O (1), where the PDF of the height takes the form P (H, t) ∼ exp(−Φ(H)/t 1/2 ), and which has required to develop new theoretical methods. The first is the weak noise theory (WNT), which has allowed to obtain numerically the large deviation function Φ(H), and analytically its tails for large |H| for a variety of initial conditions [16][17][18][19][20][21][22]. The second method uses the known exact solutions of the KPZ equation [23][24][25][26][27][28][29][30], leading to exact formula for the rate function Φ(H) for arbitrary H. It has been achieved for the droplet initial condition [31] (with an impressive confirmation from high precision numerics [32]), for the Brownian initial condition [33], and for the half-space KPZ equation with droplet initial condition [34].
In this paper we develop a systematic expansion of the KPZ equation at short time, using the exact solutions. We obtain an expansion of various generating functions associated to the height H(t) to a much higher order than previous results. We focus on the droplet initial condition, with some additional results for the Brownian initial condition.
We use a direct method starting from the known Fredholm determinant representation in Section II, and the cumulant method introduced in [35] in Section III. For the droplet initial condition we find that the additional orders lead to a very good approximation of the exact Fredholm determinant, up to large time. We also obtain the large deviation function of H and the generating function of the cumulants of H to the next order in t.
There are several other outcomes of our study, some unexpected. First, a careful examination of the structure of the short time expansion allows to obtain some results at long time. This is relevant to the recent works which have studied the large deviations at long time t 1 [35][36][37][38][39]. The left tail of the PDF of H(t) was argued quite generally to take the form log P (H, t) −t 2 Φ − (H/t) for large negative fluctuations −H ∼ t when t 1.
[ 36]. In recent works the explicit expression of Φ − (z) for droplet initial condition in the full space was obtained (i) using a WKB type approximation [37] on a non local Painlevé type equation representation of the exact solution derived in [26] (ii) using Coulomb gas methods [39]. Remarkably, we obtain here a third, and completely independent derivation of this result.
The second outcome relates to random matrix theory. There has been a lot of interest in linear statistics and central limit theorems for eigenvalues of large random matrices in the Gaussian Unitary Ensemble (GUE) [40], see e.g. [41][42][43] and references therein.
Interesting connections to the 2D Gaussian free field have been demonstrated, see e.g. [1,2,44]. There are not so many results concerning the linear statistics near the edge of the GUE, see however [3]. Remarkably, there is a connection between the KPZ equation with droplet initial condition and the Airy point process, which describes the (scaled) eigenvalues of the GUE at the edge [23][24][25][26][45][46][47]. Here we are able to use that connection to make detailed predictions for (i) the higher cumulants of the linear statistics of the Airy point process (ii) the large deviations.
The third outcome relates to non-interacting fermions in a trap at finite temperature. At zero temperature they have been well studied [40,[48][49][50][51][52]. Recently it was found that the finite temperature problem can be related to the solution of the KPZ equation with droplet initial condition [45,53]. Our results for the systematic short time expansion for KPZ can thus be transported and extended to obtain the high temperature expansion for the full counting statistics of the fermions, well beyond previous results [31,35]. Similarly our large time solution can be extended to obtain the large deviations of the number of fermions in an interval at low temperature.
Finally, we observe unexpected similarities in the expressions of large deviations for the KPZ equation on an infinite [31] or semi-infinite [34] line at short time and for TASEP with periodic [54][55][56] or open geometry [57][58][59][60]  what quantities should be related by this hypothetical duality for general initial condition.
Before we detail the aim of the paper and summarize the main results, let us introduce more explicitly the model studied here.

B. The model: KPZ equation and initial conditions
In this paper we consider the Ito solution Z(x, t) of the stochastic heat equation Exact solutions have been found for several initial conditions, notably flat, droplet and stationary [23][24][25][26][27][28][29][30], and, remarkably, can be expressed using Fredholm determinants or Pfaffians. The typical behavior of the KPZ height fluctuations has been obtained from them, and the scaled PDF of h(x, t) converges in the long time limit to the so-called Tracy Widom distributions (i.e. the distributions of the largest eigenvalues of standard Gaussian random matrix ensembles).
We study in this paper the short time expansion of the one-point statistics for a related height function H(t). We consider two types of initial conditions, for which exact solutions are known: 1. The sharp wedge initial condition, corresponding to droplet growth, defined for the stochastic heat equation as Z(x, 0) = δ(x). The first moment is normalized as . For this initial condition, we define the shifted height H as The statistics of H(x, t) is known to be independent from x. We can thus focus on x = 0 and consider only H(t) = H(0, t).
2. The Brownian initial condition with drift w, defined as h(x, 0) = B(x) − w|x|, where B is a unit double-sided Brownian motion with B(0) = 0. The case w = 0 + corresponds to the invariant measure of the KPZ equation, and is called the stationary initial condition. We will be interested in the height at point x = 0, and we define the shifted height H(t) = h(0, t) + t 12 .

C. Aim and main results
Our aim in this paper is twofold. First we construct a systematic short time expansion for the solutions to the KPZ equation in the cases where they can be obtained exactly in terms of Fredholm determinants, namely for the droplet and Brownian initial condition.
Remarkably, it will also allow us to obtain results at long time in the droplet case. Second, we use the connection between the droplet initial condition solution to the Airy point process, and generalize our calculation to obtain exact results for the linear statistics at the edge of the GUE and for the counting statistic of trapped fermions.

KPZ equation
In this paper we obtain the short time expansion for a doubly exponential generating function Q t (σ). For each initial condition, Q t (σ) is defined in accordance with the form of the exact solution. For the droplet initial condition, we define Q t (σ) as the expectation over the KPZ evolution, Using known exact results summarized in the next section, we derive the new representation (49) for Q t (σ), from which the short time expansion can be performed systematically for σ < 1. At leading orders in t, we obtain The expansion up to order t 3 is given in (61), and a conjecture for the form of the expansion to all orders in (63). The leading order O(t −1/2 ) recovers the result of [31].
Comparison with the exact Fredholm determinant representation (21) below, numerically evaluated using Bornemann's method [14], shows a very good agreement. The plots of Figure 1 (see especially the insets for t = 10) indicate however that the expansion (5) has only an asymptotic nature: adding more terms ∝ t m in the expansion only reduces the difference to the exact value (21) for small enough orders m ≤ n(t, σ), before the expansion begins to depart from the exact value when adding the orders ∝ t m with m > n(t, σ) of the expansion. Equivalently, for fixed values t, σ, the short time expansion can only approximate log Q t (σ) within a fixed range (t, σ) > 0. As usual for asymptotic series, n(t, σ) → ∞ and (t, σ) → 0 when t → 0 for fixed σ. Additionally, the plots in Figure 1 also suggest that n(t, σ) → ∞ and (t, σ) → 0 when σ → −∞ with fixed t.
The point t = 0 thus corresponds to an essential singularity of the function Q t (σ), and the expansion of Q t (σ) around this point has a radius of convergence equal to zero (i.e. there is no t = 0 such that the full perturbative series (5) converges to the exact value (21) of Q t (σ)). However, the expansion (5), whose summation is equal to the analytic part in the variable t 1/2 of Q t (σ) with essential singularities at t = 0 removed, does have a non-zero radius of convergence τ (σ) in the variable t. A numerical evaluation of the first coefficients of the short time expansion for various values of σ suggest that τ (σ) is uniformly bounded from below away from σ = 1, but converges to zero when σ → 1 − . This is supported by conjecture (63), which implies τ (σ) For the Brownian initial condition, Q t (σ) is defined as the expectation value over the KPZ evolution, the Brownian motion B characterizing the initial condition, and an additional random variable χ independent of H with probability density p(χ) = e −2wχ−e −χ /Γ(2w), Using the cumulant method introduced in [34,35] we obtain for fixedw = w √ t the short time expansion The leading order agrees with the result obtained in [33]. The limitw = 0 relevant for stationary KPZ is discussed in Section III C 4. In addition we also study in Section II D 2 another generating function, the exponential generating function F t (s), whose short time expansion is related to that of Q t (σ). For droplet initial condition, we define and obtain for s > 0 the short time expansion where σ is related to s as (for s ≥ −ζ(3/2)/ √ 4π, see Section II D 2) From (9) we can obtain the short time expansion of the cumulants E KPZ H(t) q c of the height, see Eq. (95) and Table I which agree with the values obtained in [24] (see Eqs. (11) and (12) there). In principle from our result for Q t (σ) to order t 3 one can obtain F t (s) to the same order. Since it is tedious we give here explicitly only the first three orders (the order O( √ t) is displayed in (90)).
We also obtain the corrections to the short time large deviation expression for P (H, t) in the form with the function Φ − (z) given in (140). This recovers, by a completely different method, exactly the result of [37] using a non-local Painleve equation method and of [39] using a Coulomb gas method. We have not been able to obtain the equivalent result for the stationary initial condition but we are able (see Section III C 5) to obtain the two leading orders of the rate function Φ stat − (z) in the large |z| expansion, which is consistent with the conjecture Φ stat [39].

Linear statistics of the Airy point process (edge of GUE)
Linear statistics of the GUE adresses the evaluation of averages of the type for suitably chosen functions F (λ), where the λ i are the eigenvalues of a N × N GUE random matrix. Here we will be interested in the edge of the GUE in the large N limit, which is described by the so-called Airy point process [15].
Normalizing the GUE measure at large N to a support [−2, 2] we recall that the Airy point process is the limit point process {a i } obtained by writing as N → ∞, where the eigenvalues are ordered as λ 1 < . . . < λ N . Let us denote equivalently by E Ai [. . . ] or · · · averages over the Airy point process. Introducing µ(a) = i δ(a − a i ) its empirical measure, we recall that its mean density is given by µ(a) = K Ai (a, a) in terms of the Airy kernel defined in Eq. (23). It decays (stretched) exponentially for a → +∞ and behaves as 1 π √ −a for a → −∞, leading to an accumulation of the a i for large negative values, Remarkably, the KPZ problem and the linear statistics of the Airy point process are related. The exact solution for the generating function (4) of the KPZ equation with droplet initial condition [23][24][25][26] can be written as an average over the Airy point process [46] (a type of relations which appear in a more general setting, see [47]) which also appears in the context of fermions [45] (see below).
We exploit this connection in Section V to study the linear statistics for the Airy point process, namely averages over the Airy point process of the type E Ai [exp( +∞ i=1 f (a i ))]. Since explicit expressions are difficult to obtain for arbitrary function f (x), we focus on the following two problems: • We calculate the following average in an expansion for small value of an arbitrary for a large class of functions f , which allows us to obtain the cumulants of the scaled empirical measure µ(at −1/3 ) of the Airy point process, in an expansion in small t, i.e. in an expansion from at −1/3 −1 the matching region with the bulk, and towards the edge. The second cumulant is given, to leading order in small t, by the formula (204) where H is the integrated empirical measure (189). We show that it matches the result in the bulk of the GUE spectrum obtained from the Gaussian free field correspondence by Borodin and Ferrari [1,2]. Here we obtain systematic corrections to the Gaussian free field (higher cumulants, expansion towards the edge), see results in Section V A 3. This also extends a classical result of Basor and Widom to a much higher order [3].
• We calculate the following average in an expansion for large t This is a large deviation result, with a rate function F(φ) which we obtain explicitly for a class of functions φ. It is an extension of the large time large deviation formula for KPZ with droplet initial condition (12). The formula are given in Section V B.
Both results, including the KPZ results for the droplet initial condition mentioned above, are obtained in a single framework by extending the calculation of the generating function (4) to where from (14) one sees that (4) is recovered for the choice g(x) = g KPZ (x) := − log(1 − x) = Li 1 (x) and β = 1. We have definedĝ t,σ (a) = g(σe t 1/3 a ). The second identity in (17) holds from well known properties of the Airy point process and generalises to any determinantal point process with associated kernel K and any function g (see Refs. [46,[61][62][63] for reviews on determinantal point processes.

Counting statistics of trapped fermions
In Section VI we consider N non-interacting fermions in an harmonic trap. Near the edge of the Fermi gas, x edge , the fermion density (of Wigner semi-circle mean shape) vanishes. There are however mesoscopic quantum fluctuations on distances of order w N = N −1/6 / √ 2 near the edge, and thermal fluctuations are important in the temperature [45,53]. Eq. (14) allows to relate this finite T fermion problem to the finite time KPZ solution with droplet initial conditions and to extend the KPZ results to the fermions [31,45].
Here we study the PDF of N (s), the number of fermions in the interval ξ i = b the higher orders are obtained in (238). Next we obtain explicit the small b expansion formula for the cumulants N (s) p c in the region s = O(1) where N (s) is large, see Eq.
(243). Our high temperature results extend those of [31] to higher orders.
Finally, we study the low T region, large b. There is an interesting large deviation regime for the quantity N = N (s = zb 3 ) for fixed z < 0. The typical value of N is given by the semi-circle estimate N typ = 2 3π (−z) 3/2 b 3 1, i.e. it is large. However N can fluctuate on the same scale b 3 , and its PDF takes the large deviation form at large b where (12)  with Φ(z,λ) = Φ − (z) forλ > −z and leads to formula (257). We obtain explicitly the leading terms for F (ν) in an expansion at large negative z with F 0 (ν) = 1 − 5 2ν + 3 2ν 5/3 withν = N /N typ , and the higher corrections F n (ν) are discussed there.

D. Known exact solutions for the KPZ equation
Let us recall here some details about the solution of the KPZ equation for all times t with droplet and Brownian initial conditions which will be useful in the rest of the paper.

Droplet initial condition
The moment generating function (4) of e H is given by the Fredholm determinant The kernel of the integral operator M t,σ , equal to is the product of a Fermi factor and of the Airy kernel

Brownian initial condition
For the Brownian initial condition, the exact solution is written in terms of an additional random variable χ independent of H, with probability density p(χ) = e −2wχ−e −χ /Γ(2w). The moment generating function of e H(t)+χ is again a Fredholm determinant [28][29][30], The kernel of the integral operator M Γ t,σ , equal to is the product of a Fermi factor and of the deformed Airy kernel where the deformed Airy function is equal to where Γ is the Gamma function and ∈ [0, Re(d/b)) due to the pole of the Γ function.
Remark I.1. The operators on L 2 (R) with kernels K Ai and K Ai,Γ can be written as The projector P 0 onto R + has kernel P 0 (x, y) = Θ(x)δ(x − y) with Θ the Heaviside step function, and Ai and Ai Γ Γ are operators on L 2 (R) with respective kernels II. DIRECT SHORT TIME EXPANSION FOR KPZ WITH DROPLET

INITIAL CONDITION
In this section, we consider the doubly exponential generating function of the height Q t (σ) defined in (4) for the KPZ equation with sharp wedge initial condition. After some manipulations on the exact solution (21), we obtain the alternative expression (49), from which the short time expansion of log Q t (σ) can be performed in a systematic way. The expansion up to order t 3 is given in (61), (62). At the end of the section, we also consider the generating function of the height cumulants F t (σ) defined in (8), and the PDF of the height P (H, t).
A. Doubly exponential generating function of the height Q t,β (σ) As explained in the introduction it is advantageous for the short time limit to consider the slightly more general generating function log Q t,β (σ) = q t,β (σ) defined as where {a j } forms the Airy point process, γ = t −1/3 , β is a scalar parameter which we will use below for bookkeeping purpose (and is set to β = 1 in all applications). Here we consider general functions g with g(0) = 0 which is mandatory for convergence, another condition being g(−∞) < 0. The results for the short time expansion of the KPZ equation with droplet initial condition, for the generating function Q t (σ) defined in (4), are recovered for the special choice The expectation value over the Airy point process has the Fredholm determinant ex- We recall thatĝ t,σ (a) = g(σe t 1/3 a ). Here and below cyclicity on the variables u j is assumed, u j+m ≡ u j . It is convenient for the following to consider instead the derivative q t,β (σ). This leads to a sum over k = 1, . . . , m of terms where the factor exp βg(σe u j /γ ) is replaced by ∂ σ exp βg(σe u j /γ ) . By cyclicity of the product over j, one can make the derivative act only on the factor with k = 1 after renaming the variables u j , and the factor 1/m of the Fredholm expansion cancels. Using then one finds after partial integration on u 1 The derivative over u 1 gives a sum of two terms, containing respectively a factor ∂ u 1 K Ai (u 1 , u 2 ) and a factor ∂ u 1 K Ai (u m , u 1 ). Shifting the indices of the variables u j by 1 in the former, the integrand can be factorized as (. . .) × (∂ um K Ai (u m , u 1 ) + ∂ u 1 K Ai (u m , u 1 )).
Writing the Airy kernels explicitly as (23) and expanding the pre-function as the integration over the variables u j can be performed using the Airy propagator √ 4πn (37) After the change of variables z j → z j / √ γ, one finds (we recall that m j=1 a n j ,β σ n j e t 12 n 3 We note that for β = 1 and g(x) = Li 1 (x), (38) is essentially identical to some intermediate expression written in the derivation [24,25] of (21) by the replica method: between (32) and (38), we have essentially just done part of the replica calculation backwards.
The change of variables z j → z j − √ t( j =1 n )( m =j+1 n ) cancels the linear terms in the z j inside the exponentials. Using the identity valid for any m ∈ N * and {n j } j∈ [1,m] a set of integers, it leads to The factor e t 12 ( m j=1 n j ) 3 can be conveniently written as the action of the operator e t 12 (σ∂σ) 3 on σ m j=1 n j . One has .
For any function f such that F (ε) = ∞ ε dz f (z) converges, one can write formally This identity can be used to perform the short time expansion of q t,β (σ). Each integral over z j gives two terms, corresponding respectively to the integral from 0 to +∞ and to the action of the corresponding operator e ε∂z j −1 This leads to The various n j are still coupled. In order to disentangle them, we replace σ n j by σ n j j and interpret the other n j as σ j ∂ σ j . One finds The multiple integral over z k j +1 , . . . , z k j+1 −1 is very similar for each j = 0, . . . , p. We After renaming the z k j to z j , the generating function becomes , we observe that the action of any power of the operator k i+1 =k i +1 σ ∂ σ on an arbitrary function L(σ k i +1 , . . . , σ k i+1 ) is equal, after taking all the σ equal to σ, to the action of the same power of the operator σ∂ σ on L(σ, . . . , σ). Thus, all the parameters σ inside a given function L in q t,β (σ) can be set equal from the beginning. Replacing σ k j +1 , . . . , σ k j+1 by σ j and introducing the function Performing the change of variables k 1 → k 1 + k 0 then k 2 → k 2 + k 1 + k 0 , and so we finally arrive at B. Algorithm for the short time expansion of Q t,β (σ) In this section, we explain how the short time expansion of (49) can be performed systematically to arbitrary order in t. This leads to the short time expansion (61) for log Q t (σ). The algorithm giving the expansion is based on the following identities verified by the functions L β (σ, a, b) defined in (48): These identities are proved in Appendix B 3. From (49), each term in the short time expan- In terms of the L i,j,k , the identities (50) -(53) imply At each order in t in (49), the index k of the variables L i,j,k can be set to zero everywhere after applying recursively (56). Then, using (57), the index j can be reduced as well, as long as i ≥ 1. Because of this constraint on i, it is not guaranteed that (57) will be sufficient in order to make j = 0 everywhere. In practice, we observed that at least up to order t 3 , using repeatedly (57) does eliminate all variables L i,j,k with either j > 0 or k > 0: only variables of the form L i,0,0 remain. For the first orders in t, we find where L i = L i,0,0 has from (54) the explicit expression Computing explicitly the action of the operator e t 12 (σ∂σ) 3 and integrating with respect to σ, we observe empirically that the integral at each order can be performed explicitly, except the one at order t 0 . The term at order t 0 is given by We finally obtain our main result for the expansion of q t,β (σ) defined in (32) for an where the L i are defined in (59).
Application to KPZ. We specify the function g( Then, the short time expansion of the generating function of the height Q t (σ) for sharp wedge initial condition defined in (4) is given in terms of (61) by log Q t (σ) = q t,1 (σ) with The first terms of the expansion are written more explicitly in (5).

C. Conjecture for the form of the general term
We have not been able to obtain precise analytical expressions for arbitrary high orders in t of log Q t,β (σ). The expansion (61) up to order t 3 however leads us to the following conjecture for the form of the general term of the expansion.
Conjecture II.1. For σ < 1, the short time expansion of log Q t,β (σ) has the form The notation n is a shorthand for (n 1 , . . . , n r+2−2q ). The coefficients c r,q (n) are positive integers, and S(n) is a symmetry factor equal to S(n) = m 1 ! . . . m k ! when the n j 's take k distinct values with multiplicities m 1 , . . . , m k .
The coefficients c r,q (n) extracted from (61) are listed in the Appendix A. The fact that all the coefficients c r,q (n) appear to be positive integers suggests the existence of a combinatorial interpretation. We have not managed to guess exact expressions for the coefficients c r,q (n), except for c r,0 (n) = (r − 1)!, which is easily spotted in Table IV, and is crucial for the considerations in Section III B 3.
In the special case of the KPZ equation with sharp wedge initial condition, where L i is given by (62), the radius of convergence of the short time expansion can be extracted from (63) in the limits σ → 1 and σ → −∞. When σ → 1, L i (63) is at leading order proportional to , and only q = 0 contributes to (63) at leading order. Each term contributing to the coefficient of t r/2 is at leading order proportional to (log(−σ)) 2−r/2 , and the radius of convergence is thus proportional to log(−σ) when σ → −∞. The resummation (140) below gives more precisely a radius of convergence in the variable t of order π 2 log(−σ).

D. Height distribution P (H, t) and generating function F t (s)
In this section, we consider the probability distribution function P (H, t) of the KPZ height field with droplet initial condition, and the associated moment generating function F t (s) defined in (8). By consistency with our results, as sketched below, they admit the following short time expansion The leading order Φ(H) (resp. λ(s)), already obtained in [31], agrees with our present results. Here we explicitly determine the next order corrections Φ 0 (resp. µ 0 ), but the method can be extended to obtain iteratively the higher orders.
Our starting point is the short time expansion of the generating function of e H where we have defined the notations and similar notations for the higher order functions which can be read from Eqs. (61) and (62) . Since here we only consider the leading order correction, we drop all terms above Φ 0 , µ 0 and Ψ 0 . All equalities will then have to be understood with an O( √ t) error term. In the rest of the section, we first evaluate the height distribution and then the generating function of H.

Short time expansion of P (H, t)
The l.h.s of (66) can be written as an integral involving the height distribution of H, Inserting the expansion (64), this leads to At leading order in t, doing a saddle point evaluation, one recovers the result of [31], extended in [34], which we express in a parametric form (H * indicates the saddle point) The next term in the short time expansion results from the Gaussian integration around the saddle point (once again the overall error is of order O( √ t) and does not contribute Using the following identity (obtained by differentiation of (70)) for k = 2 one simplifies the prefactor of the exponential as (the minus sign is due to the fact that Ψ is strictly concave [34] and ensures that the saddle point is a maximum of the exponential) We find by identification We now summarize the above results for the first orders of the distribution of H (dropping the * ).

Explicit formula
Using the exact expressions of Ψ and Ψ 0 of (67), and the analysis performed in [31,34], we obtain the parametric representation for the two leading terms of the distribution of H for the droplet initial condition We symbolically write these continuations as Li → Li + ∆. The interpretation of this shift ∆ comes from the fact that the polylogarithms have a branch cut at σ = 1 and one can extend them in an upper Riemann sheet by adding the contribution of the residue at the branching point which is ∆. Taking into account the shift ∆, the solution for H > H c is given by the following parametric system (where the range of σ is now ]0, 1]) where K is a constant given by It is shown in the Appendix D that this is the proper continuation to H > H c , and that  Inserting in the second line we find Insering in the third line we find • The asymptotics near H = +∞ (i.e. σ = 0) are as follows. Let us denote σ = e −s with s → +∞. Here we must use the analytical continuation for H > H c . The first equation in (77) gives Inserting in the second line of (77) we find Insering in the third line we find

Short time expansion of F t (s)
To obtain the different contributions to F t (s), we express the middle term of (65) as an integral with the height distribution of H.
Inserting both large deviation ansatz, we seek the following identification The leading order in t is obtained by a saddle point evaluation and the first correction is obtained by the Gaussian integration close to the saddle point. After identification, the final result is expressed parametrically in terms of H or equivalently in terms of σ (using Eqs. (70) and (72)) where we recall that Φ (H) > 0 as Φ is convex. Higher orders can be obtained by expanding around the saddle point. We find Using the exact expressions of Ψ and Ψ 0 of (67) one obtains the final result. The parametric representation of the two leading terms of the moment generating function of H is The representation is valid for s ≥ − ζ(3/2) √ 4π and for σ in ] − ∞, 1]. To obtain the representation for values of s ≤ − ζ(3/2) √ 4π , one extends the parametric system as (the range of σ is now ]0, 1]) The continuation of µ 0 (s) can be obtained using similar methods as in the previous Section (see also Appendix D).
From this expansion, one can obtain the first few cumulants of H by expanding the system (91) around σ = 0 and compare them with the ones obtained in [24] Eqs (11) and (12). Up to O(σ 4 ), the system (91) yields The inversion of this system up to O(s 4 ) reads By definition, the cumulants of H are given by For q = 1, 2, 3 the leading orders are given in Table I which confirms the values obtained in [24] Eqs. (11) and (12) for the cumulants of H.

Cumulant
Leading orders In this section we study the cumulants (defined below). First we extend the cumulant expansion method introduced in [34,35] to obtain a few terms in a systematic short time expansion of the moment generating function. This method is more versatile that the one of the previous section, and allows to treat also the Brownian initial condition, in addition to the droplet initial condition. For the latter, it allows to obtain the few lowest orders quite easily, and agree with the results of the previous section. However, since it becomes quickly quite challenging we have not attempted the systematic study of the higher orders.
In a second part we list the results for the cumulants from the direct method. We study their (conjectured) general structure which then allows us to also obtain large time results.

A. Presentation of the cumulant method
Let us describe the method. As recalled in Section I D, the generating function for the KPZ equation for both initial conditions can be expressed using a Fredohlm determinant which involves a generic kernel K, either (21) with kernel K = K Ai in (22) for the droplet initial condition, or (24) with kernel K = K Ai,Γ in (25) for the Brownian initial condition, multiplied by a weight function which here we will take quite general, see (22) and (25) for KPZ.
We use the fact that, for any kernel K and function g, we can expand the following Fredholm determinant in powers of g as where for applications we will setĝ(a) =ĝ t,σ (a) = g(σe t 1/3 a ). This provides an expansion in cumulants for the more general generating function of Q t (σ) defined in (17) in the case where K is the Airy kernel. The first three cumulants are given by and the formula for the general cumulant is This expansion is known as the cumulant expansion see [34,35]   the droplet (i.e. narrow wedge) initial condition, κ n , up to n = 5.

Introduction of the propagators
We have seen in Section I D that for both initial conditions, the kernels K Ai and K Ai,Γ can be factorized into a product of two (deformed) Airy operators. Since the cumulants are expressed in terms of traces involving these operators, it is useful to use the cyclicity of the trace and study the operators Aiĝ Ai : L 2 (R) → L 2 (R) defined as and similarly for Ai Γ Γ . Upon Taylor expandingĝ t,σ asĝ t,σ (v) = +∞ p=1 σ p p! g (p) (0)e pv/γ , we introduce the Airy and deformed Airy propagators as which represents the propagator of a Brownian particle in a linear potential, and which is proved in Appendix. Hence we have and the same for Ai Γ Γ . We can also express the operators involving powers ofĝ and similarly for higher powers. Note that these propagators satisfy the reproducing property Property III.1 (Reproducing property).

Expression of the cumulants and diagrammatic representation
Using the reproducing property and the propagators we now obtain explicit formula for the first three cumulants in terms of the propagators and the projector P 0 on R + .
These formula have also a diagrammatic interpretation which we provide here. In each case, we use some algebra to reduce the number of terms. Since each term is superficially of a lower order in 1/γ = t 1/3 than the total, this also allows to automatically perform the cancellations.

• First cumulant
From (97) and (102), using the cyclicity of the trace we obtain which has the following diagrammatic representation. Here we integrate the variables over the vertical thick lines. The interpretation of this graph is that we sum all paths starting and ending at the same point on the positive axis (i.e. bridges) From the formulae (97), (102) and (103), using again the cyclicity of the trace together with the reproducibility (104) we obtain the second cumulant as which has the following diagrammatic representation. Here the bridge must cross the zero line at least once.
From the second line in (97), (102), (103) and its generalization, using the cyclicity of the trace together with the reproducibility (104) we obtain the third cumulant as where we have used the (p 1 , p 2 , p 3 ) permutation invariance. From the last line we see that the third cumulant is given by the antisymmetrization of the first trace, since in the second trace all projectors are complementary, so that where . It can be written diagrammatically as The above algebra can be extended to any order, and is valid for both initial conditions, upon replacing the propagator G p by their expressions due to the particular form of the two kernels (100) and (101). We now start the explicit calculations for the droplet initial condition, the Brownian one is performed afterwards.
B. Cumulants associated to the Airy kernel and application to KPZ with droplet initial condition 1. Evaluation of the first three cumulants

• First cumulant evaluation
To calculate the first cumulant at any order, we start with After summation one obtains the first cumulant at any order in t.
Application to KPZ III.2. Taking g(x) = g KPZ (x) = Li 1 (x), the first cumulant for droplet initial condition reads We provide two equivalent expressions for the first cumulant (see Appendix for the derivation) and • Second cumulant evaluation To obtain an exact expression for the second cumulant, we start by calculating the trace involved in (106) This integral can be explicitly performed using the change of variable (note that there is a 1/2 Jacobian) which leads for the trace One can proceed to a general summation to obtain κ 2 (g) as a double series.
Keeping the two first terms of the expansion of the error function, we give the two first contributions to κ 2 (g) where we recall the notation L i = (σ∂ σ ) i +∞ −∞ dp 2π g(σe −p 2 ).
Remark III.4. It is possible to obtain an exact expression for κ 2 (g) to all orders.
Starting instead from (106) and (114) and inserting the form of the propagator given in the last equation in (100), we can perform the summation over p 1 and p 2 , use the change of variable (115) and a rescaling of k to obtain • Third cumulant evaluation Let us first provide an exact formula for the summand in the third cumulant ex-pression (108), in the form of an integral.
A Tr( We rescale the arguments so that the Gaussian exponential do not have any γ dependence.
A Tr( Until now this expression is exact. We have not attempted to calculate explicitly this integral. We now provide its small time expansion up to the second order, leading to a perturbative formula for the third cumulant. Expanding the hyperbolic sine up to the third order, we obtain where and The calculation of both integrals is non trivial but simply leads a homogeneous polynomial in the p that is symmetric, hence easy to identify using Mathematica. The last equality has to be taken modulo the permutation of p 1 , p 2 , p 3 denoted by the symmetry group S 3 . We therefore evaluate the trace up to order O(t 5/2 ) as We summarize the two contributions we obtained for the third cumulant where we recall the notation L i = (σ∂ σ ) i +∞ −∞ dp 2π g(σe −p 2 ).
Application to KPZ III.5. Taking g KPZ (x) = Li 1 (x), the third cumuland reads Remark III.6. It is possible to obtain an exact (although a bit formal) expression for κ 3 (g) to all orders, see Appendix (C 2).
• Summary: first three terms in the cumulant expansion for droplet KPZ Collecting the contributions above we obtain, for the droplet initial condition, the following expansion This expression contains the first three orders in the small time expansion. The leading order recovers the result of [31]. It is interesting to note that the degree in terms of polylogarithm indicates from which cumulant each term originates (i.e. a product of n polylogarithms comes from the n-th cumulant). The fourth order, i.e. the order t is not complete as we miss the leading term from the fourth cumulant, not calculated here. By comparing with Eq. (5) we see that the two series agree to this order. Although it allows to obtain the few leading orders painlessly, the systematics of this cumulant method remains to be improved to match the method of Section II. We now turn to the cumulant calculation using the latter method.

Cumulants by the direct expansion method
In Section II we introduced the parameter β as a counting device. From the definition of the cumulants in Eq. (98) it is easy to see that it exactly counts the order of the cumulant. Hence we can now list the results for the cumulants from the direct method of Section II. The first cumulants κ n (g) = (∂ n β q t,β (σ)) |β=0 are then equal to where the L j are defined in (59) (setting β = 1) for general g(x) and in (62) for KPZ with droplet initial condition. Note that an exact formula to all orders exists for the first cumulant κ 1 , see Eq. (110).

Structure of the cumulant series in the large time limit
Since we are studying formally an expansion in t, but for arbitrary fixed σ, it turns out that one can in fact obtain some results for arbitrary t and even for large t. Obtaining such results assumes that the observed structure of the series in t when σ → −∞ holds to arbitrary order (and in a non perturbative sense). Examination of our results (129) for the cumulants, together with the conjecture formulated in Section II C leads to the following guess for the structure of the cumulant expansion where the leading order is obtained from the conjecture c r,0 (n) = (r − 1)! of Section II C, for n ≥ 1. Indeed it holds for n = 1 since L 1 = (σ∂ σ ) 2 L −1 and κ 1 = t −1/2 L −1 and for The Eq. (131) can now be studied at fixed t but in the limit σ → −∞. In that limit the first term in (131), t n 2 −1 κ 0 n (σ), is the dominant one, which gives the leading asymptotics of Q t (σ) for σ → −∞ for any t. Indeed we see from (129) and the above conjecture that the term t Remarkably, the structure of the series (131) is such that the large time limit is also controled by the term t n 2 −1 κ 0 n (σ). This can be seen as follows. Let us choose σ = −e −zt , with fixed z < 0. From the conjecture above we can write in a symbolic form the general term of (131) as where here we just count the degree of homogeneity in derivatives and function L 1 . Since the p dependence in the power of t behaves t −2p we see that p = 0 is the leading term for each fixed n in the large t limit.
Until now our considerations where valid for general function g(x). We now carry the explicit summation of all cumulants in the limit t → +∞ at fixed z, in the case of KPZ with droplet initial condition.
Application to KPZ III.7. Using (132) and (133) we obtain for each cumulant and we note that all cumulants are proportional to t 2 , which results from our choice σ = −e −zt , it is the only choice at large time which allows all cumulants to have the same homogeneity in the time variable. The summation over n can then be explicitly performed, and we find that at large time where Φ − (z), z < 0, is precisely the function obtained by completely different methods in [37] using a WKB analysis of a non-local Painleve equation and in [39] using a Coulomb gas method This is a large deviation result for the left tail H < 0 of the probability distribution of the KPZ height at large time (12), since for t 1 We recall that it predicts a crossover between the cubic tail of the Tracy Widom distribution (matching the typical fluctuations regime) and Φ − (z) z→0 − 1 12 |z| 3 and the |H| 5/2 outer left tail Φ − (z) z→−∞

C. Cumulants for KPZ with Brownian initial condition
We proceed as in the previous section. For the Brownian initial condition the propagator is given by Eq. (101). We know, from the previous work on the Brownian initial condition [33], where the leading order at small time was obtained, that one must scale to obtain a well defined short time limit with a continuously varying scaled drift parameter w. We will study this limit here. Note also that it was found that the proper generating function is which amounts to shift the height field by log t. We now work with g = g KPZ .

First cumulant evaluation
We recall from section III A 2 that the first cumulant is given by The trace of the propagator (101) reads One can simplify this trace by proceeding to the integration w.r.t r, further rescale k by √ γ and introduce the rescaled driftw. This leads to an expression for the first cumulant (144) valid at all orders in time.
The summation cannot be done per se, and we therefore employ Stirling's approximation to obtain an expansion of the four Γ functions in powers of γ −3/2 = √ t.
This allows to obtain the first two leading orders in the large γ expansion of the first Remark III.8 (Zero drift limit of the first cumulant). It was shown in [33] that the leading order of κ 1 withw = 0 is well defined and can be written as a series of From this the rate function Φ(H) was obtained for the stationary casew = 0. To the next order, the limitw = 0 is more tricky will be analyzed below.

Second cumulant evaluation
We now study the second cumulant, and we will restrict to its leading order in the small time expansion. We recall the formula (106) We only need here the leading order term of the propagator (see previous section) One has, after rescaling (r 1 , r 2 ) → γ −1/2 (r 1 , r 2 ), where in the last line, to this order, we have discarded (i.e. set to unity) the term e − p 1 +p 2 γ 3/2 r 1 +r 2 2 , performed the change of variable in (115), and integrated over v. We can insert in (150) and sum over p 1 , p 2 to obtain

Summary
We can now put together the previous calculations and obtain the two leading orders for the Brownian initial condition. Recalling that γ = t −1/3 we obtain the following The leading order recovers the result of [31] in an equivalent form, indeed one has which is easily checked setting y = k 2 and performing an integration by part. The r.h.s. of (157) is precisely the result Eq.(16) of [31]. One can check that forw → +∞ one recovers the result for the droplet initial condition. The limitw → 0 is discussed in the next subsection.

Height distribution P (H, t) at short time
Let us proceed as in [33], Supp. Mat. Section 4.1, and obtain the next order in the short time expansion. We consider the leading behavior for fixedw, which implies that w =w/ √ t is large. We perform the shift χ → χ + ln( √ t), use Stirling's formula for Γ(2w) factor in the PDF of χ given below Eq. (23), and write, using the form (64) for P (H, t) One first proceeds to the saddle point on χ at fixed H and to that aim define take χ H to be the critical point, such that Ξ (χ H ) = 0, then we have using the saddle point method The critical point and the second derivative are given by which leads to formula (79) in [33], namely and comparing with (155) we obtain at fixed σ with C(t) = 1/ 2π √ t. Here Ψ and Ψ 0 are given explicitly in (156). The evaluation of the second derivative gives d 2 which must be complemented with the relation between σ, H, Φ(H) in the parametric representation [33] Note that the above results are valid for H < H c (w) with and we will not attempt here to extend the analysis of [33] for Φ(H) (which involves two successive analytical continuations one exhibiting a phase transition) to obtain the expression for H > H c (w) for Φ 0 (H). It would be interesting to see the signature of the phase transition on the subleading rate function Φ 0 (H).

Stationary limit
Let us now discuss the stationary KPZ limitw → 0. The leading term Φ(H) is well defined in this limit as was discussed in [33]. Indeed, Ψ(σ) has the following limit see remark III.8, and Φ(H) is thus determined by the parametric equation for H < H c . The continuations for H > H c are obtained in [33]. We now examine the limit of the subleading rate function Φ 0 (H). It is given formally by with the same relation (168) between σ and H. We now examine Ψ 0 (σ) as given by (156) and show that the limit (170) exists, which is non-trivial. The first term in (156) behaves where the O(w) term is given by We thus see that there is a logarithmic divergence asw → 0 in the first term. It does not exactly cancel the − 1 2 ln(w) in the definition ofΦ 0 , hence we hope that the second term in (156) will cancel the remaining divergence. We now show that the cancellation does occur, i.e. the second term is − 1 2 ln(w). For this let us call F (σ,w) this second term and applyw∂w on the second term. After one integration by part we obtaiñ We are interested in the limitw → 0. For this we use the following rescaling k →wk, p →wp, u → u/w. In the limitw → 0 with fixed σ we obtaiñ which shows the cancellation.
Calculating the derivative ∂ σ F (σ,w) we obtain after integration by part and taking the limitw = 0 we obtain the derivative of the limit from which one can obtain Φ 0 (H) using the above equations.

Application to the limit t → +∞
Let us now evaluate the first two cumulants in the large t limit setting σ = −e −zt as in Section III B 3. From Eq. (148) we find The first term O(t 2 |z| 5/2 ) is the leading behavior of the rate function Φ Brownian − (z), which thus has the same leading behavior (for anyw) that Φ droplet − (z) = Φ − (z). The second term is a O(1) correction at large t, which furthermore has the finite limit z/2 asw → 0.
For the second cumulant (154), we find, replacing at large time log(1 − e −k 2 −zt k 2 +w 2 ) (−zt − k 2 ) + and rescaling k and u which recovers exactly the subleading behavior at large |z| of Φ − (z), hence is in agreement with the conjecture Φ stat (z) = Φ drop (z) based on Coulomb gas arguments [65].

IV. COMPARISON WITH STATIONARY LARGE DEVIATIONS OF TASEP
In this section, we point out unexpected similarities between short time expansions for the KPZ equation defined on an infinite or semi-infinite line and stationary large deviations of TASEP in a periodic or open domain.
A. Leading terms at short time for the KPZ equation We consider the solution of the KPZ equation on the semi-infinite line R + with sharp wedge initial condition at 0 and prescribed boundary density ∂ x h(0, t) = ρ. In both cases, the generating function Q t (σ) of (4) (with some specific definition of the shifted height H(t), see [34] ) has the short time behaviour log Q t (σ) −Ψ(σ)/ √ t. Exact formulas for ρ = −1/2, 0, +∞ [34] seem to indicate that the function Ψ only depends on whether the boundary density ρ is finite or not. For finite density (reflective wall), one has [34] Ψ which is the same as for the droplet on the full line up to a factor 2, while when ρ → +∞ (hard wall), one finds instead [34]

B. Stationary large deviations for periodic and open TASEP
The totally asymmetric simple exclusion process (TASEP) is a microscopic model of driven hard core particles which belongs to KPZ universality. We summarize here some known results for the stationary large deviations of the corresponding height field.

The first result for stationary large deviations of TASEP was obtained by Derrida and
Lebowitz [55] in the case of a system with periodic boundaries. In terms of the KPZ fixed point with periodic boundaries, which describes a random field h(x, t), x ≡ x + 1 obtained as the limit of strong non-linearity of the solution of the KPZ equation, the result can be stated in a parametric form as (see e.g. [54]) The result is the same for any initial condition. It is also independent of x by translation invariance of the stationary state.
Subleading corrections to (181) depend on the initial condition. They have been studied in [56] for special cases, see also [54]. For periodic sharp wedge initial condition, one has in particular up to exponentially small corrections in time, with σ still defined as in (182). [57], see also [58][59][60]. The results depend on the density of particles ρ a , ρ b in the left and right reservoir (with particles hopping from left to right) according to the phase diagram of the model drawn in figure   4. Only the leading term in time of the large deviations is known so far. The results below are given in terms of the random field h(x, t) characterizing the KPZ fixed point normalized in the same way as in the periodic case, and with ∂ x h(0, t) and ∂ x h(1, t) prescribed by the boundary densities. Two cases are of particular interest for comparison with large deviations at short time for the KPZ equation on R + : the edge and the bulk of the maximal current phase.

Stationary large deviations for open TASEP connected to reservoirs of particles at both ends have been obtained by Lazarescu and Mallick in
At the transition between the low density and the maximal current phase, corresponding formally to ∂ x h(0, t) = −1/2 and ∂ x h(1, t) = −∞ [66,67], the generating function is essentially the same as for the periodic system, The result is the same at the transition between the high density and the maximal current phase, where ∂ x h(0, t) = +∞ and ∂ x h(1, t) = 1/2.

C. Hints of a duality for the KPZ equation in finite volume
We discuss now the similarities between large deviations at short time for the KPZ equation and at long time for TASEP. We consider separately the case of systems with and without boundaries.

No boundary
The similarity between the expression (5), (9), (10) for the KPZ equation on R with droplet initial condition at short time and the expressions (182), (183) for periodic TASEP with sharp wedge initial condition is striking. This was already noted in [31] (see Supp. Mat. Section 5) for the leading order, but here we see that the similarity persists to the next order. This similarity suggests the existence of a duality for the KPZ equation  where here the a i are just integration variables, not the points of the Airy point process.
On the other hand (188)-(190) are nothing but our generating function Q t (σ), and from (17) and (96) we see that the cumulants of the empirical measure can be extracted from the cumulants defined in Section III A, since The short time series expansion for the cumulants κ n (g) were obtained in (129), where we recall that the L i have the explicit expression Explicit comparison of (129) and (191) for a generic fonction g (i.e. for a generic f ) allows, by identification, to read off the expansion of the scaled cumulant t −n/3 µ(t −1/3 a 1 ) . . . µ(t −1/3 a n ) c in powers of t 1/2 at small t. This expansion clearly corresponds to studying each cumulant in the large negative a region of the Airy point process, i.e. starting from the matching region with the bulk of the GUE, and expanding towards the edge. This was the purpose of inserting a power t 1/3 in the argument of f in (188).
Note that from our guess (130) for the general structure of the cumulants we can conjecture that the proper form of the series expansion of the scaled cumulants is, schematically where the δ p are distributions, some of them discussed below.
Remark V.1. We note that the leading term for the first and second cumulants, i.e. κ 1 at order t −1/2 and κ 2 at order t 0 in (129), have previously been obtained by Basor and Widom [3] for arbitrary g(x), by quite different methods. Here we obtain the terms in the expansion to a much higher order. While the O(t −1/2 ) term for κ 1 readily coincide, we obtain the O(t 0 ) term of κ 2 in (129), i.e. κ 0 2 , in the form However, it can be rewritten equivalently in the form obtained in [3] as The equality is checked by Taylor expanding the function g in both r.h.s of (194) and (195) and checking that the series match. We can also observe that (195) is the leading order in t of the exact result given in (120). We will use the form (195) below, as it is more convenient for our present purpose.
It is straightforward to identify the leading term a well known result, since the mean density of the Airy point process must match the semi-circle density of the GUE for a → −∞. Identifying the next orders in the small t expansion is more delicate, since they only admit a distributional limit. Consider a function g(σe −x ) which vanishes fast, as well as all its derivatives, near x = 0 + . Then we can rewrite and integrate by parts. This leads to where we have used the formula ∂ a On the other hand, we know the exact result for the mean density of the Airy point process µ(a) = K Ai (a, a) = Ai (a) 2 − aAi(a) 2 . Asymptotics of the Airy functions allow indeed to recover (199), but only upon discarding terms which are fast oscillating for large negative a. Hence it is valid in the weak sense and only for smooth test functions, the general formula being (196).

Second cumulant to leading order and the Gaussian free field
We now write the identification formula (191) to the second cumulant level, at leading order in t, that is κ 2 (g) κ 0 2 at order t 0 , using (195). Let us define h(a) = g(σe −a 2 ), h(u) = 1 2π R da e iua h(a) and recall that f (a) = g(σe a ). We rewrite (recalling that h is even) From (191) we want to identify this expression, to leading order at small t with Hence we obtain, by identification, for the second cumulant of the height field This logarithmic correlator is a particular case of the correlator of the Gaussian free field, as we now discuss. In [1,2] it is shown that the height field associated to the GUE, H GUE defined likewise by counting the number of eigenvalues above a certain level, is described upon rescaling by a Gaussian free field with a specific correlator (see also [44]). Taking Its inverse has the form Ω −1 (z) = 2 (z). The theorem of Borodin [2] states that H GUE (x, y) with λ GUE = x and y = 1 satisfies with the Gaussian free field correlator on the upper half plane with Dirichlet boundary condition We now take the limit of Borodin's formula near the edge x = 2 + a N 2/3 . The matching from the bulk going to the edge, with the edge going to the bulk, allows to identify which is perfectly consistent with our result (204).

Deeper towards the edge: beyond the Gaussian free field
As we go deeper towards the edge there are two types of corrections to the Gaussian free field. The first one are still Gaussian with corrections to the logarithmic variance of the Gaussian free field. The second arise from higher cumulants. We use repeatedly Eq.
• Let us first study the former, that is higher corrections in t to κ 2 . Using (129) it is straighforward to obtain the distribution result, for appropriate test functions f One sees that the next contributions in the second cumulant are sum of separable contributions in the variables a 1 and a 2 , and therefore the fully entangled part comes the zero-th order given by the Gaussian free field.
• Let us now study the contribution from the next cumulant, i.e. κ 3 . At leading order, from (129), we obtain the following correction from the third cumulant of the height field of the Airy process, which reads leading to the following expression for the third cumulant of the Airy point process which, remarkably is quite simple to this order, and factorized. Higher Here we generalize the large time calculation of Section III B 3, performed there in the context of droplet KPZ, to a more general large deviation linear statistics problem for the Airy point process. The general aim is to study, in the large time limit and calculate the rate functional F(φ), for a given set of functions φ. We start with the following choice of functions parameterized by z < 0 and γ which for γ = 1 correspond to the calculation performed in Section III B 3. We will adopt the same notation and denote the rate function F(φ) = Φ − (z). To carry out the calculation we first find a suitable function g so we can apply the considerations of Section Note that there is a time dependence in g, equivalently a factor t 1−γ in the parameter β.
We aim to evaluate (212) by applying the following result from Section III B 3, i.e. with where we have used again only the leading term in each cumulant (the . . . The choice v = −zt is made such that the main contribution of all cumulants is now homogeneous to t 2 and can be therefore summed together. We can thus identify the function Φ − (z) as the following series, summed over all cumulants Defining we obtain the large deviation rate function We give in Table III a few examples (in increasing order in γ) for which the summation of the series is easy. The resulting function Φ − (z) is found to be positive as required.
Remarkably, these results exactly agree with calculations using a completely different method related to the Coulomb gas [65]. This comes in strong support of the validity of the "conjectures" discussed above for the present method. We can generalize to a larger set of functions functions φ(x) by choosing, with σ = −e −zt with δ i > 0 and c i > 0. For this class of functions we find It remains an open problem whether this formula can be extended to a larger class of function φ(x).

FERMIONS AT FINITE TEMPERATURE
Here we apply the connection between non-interacting fermions near the edge of a smooth trap at finite temperature and the solution of the KPZ equation with droplet initial condition for arbitrary time [45,53]. The short time expansion for KPZ corresponds to the high temperature expansion for the fermions. The leading order at high temperature was studied in Ref. [31] and here we obtain many higher orders. In addition we also study some properties in the low temperature which corresponds to the large time limit.

A. Edge fermions
Let us summarize the problem, for more details see [45,53] (see also [35]  We consider the grand canonical ensemble, where the mean total number of fermions N , is large. However from the equivalence between ensembles for local observables (as considered here) we expect the conclusions to hold also for the canonical ensemble at fixed N [53] (see Ref. [68] for cases where deviations occur). It was shown in [45,53] that if one defines the reduced fermion coordinates where w N = N −1/6 / √ 2, then the set ofξ i form a determinantal point process with an associated kernel which, in the limit N, T → +∞ with b fixed, takes the form To study the high temperature expansion, it is more convenient to define as in [31] the reduced fermion coordinates ξ i = bξ i , [69], i.e. where Hence we can use our results for Q t (σ) with the choice g(x) → g λ (x) and obtain immediately the Laplace transform (228). From (61) we obtain the high temperature expansion where the . . . are all terms beyond O(b 3/2 ) in (61) obtained using the replacements As discussed in [31] the leading term for small b (high temperature) shows that the CDF of ξ max is peaked around the typical value ξ typ = log( 1 b 3/2 √ 4π ) and has the form of a Gumbel distribution. Indeed keeping only the leading term using that Li a (y) ∼ y at small y. Now we can use (233) and obtain the systematic corrections away from the Gumbel distribution. One finds log P(ξ max − ξ typ <ŝ) = −e −ŝ + 1 2 Note that to each order in b there is are contributions from the various cumulants. Note also that the expansion (237) works at fixedŝ but fails whenŝ ∼ −ξ typ ∼ 3 2 ln b, i.e. when ξ max is of order one or negative. This is the region s = O(1) which is the region of large deviations for the PDF of ξ max as discussed below.
Similarly one can study the PDF ofN (ŝ) = N (s = ξ typ +ŝ), i.e the number fluctuations around the typical position of the rightmost fermion. One finds which recovers (237) for λ = +∞. The leading term at high temperature b → 0 corresponds to the Poisson distribution pn(n) =n n n! e −n which leads to log e −λn = −n(1 − e −λ ) with a meann = e −ŝ . This reflects the fact that the positions of the fermions become independent random variables at high temperature. The next orders quantify the deviations from the Poisson distribution (it can formally be written as a convolution of distributions N = qn where n is Poisson and q = 1, 2, 3.. with however non-positive or random parametersn). The first cumulants (mean and variance) are As we show below, and obtained in [31], in the region s ∼ O(1) the mean number of fermions in [s, +∞[ is large, and behaves at high temperature as O(b −3/2 ), more precisely which is only the leading term. Here we obtain the systematic small b expansion of the cumulants in that region s ∼ O (1). Since N (s) is typically large, the probability of N (s) = 0 i.e. of ξ max < s is small, and accordingly, (233) also contains the information about the large deviations of the CDF of ξ max at high temperature. The leading term exhibits a |s| 5/2 tail which was discussed in [31]. The formula (233) with e −λ → 0 thus contains all the systematic corrections in the small b expansion to the rate function of these large deviations (which we will not further discuss).
The cumulants of N (s) in an expansion at small b and fixed s can be obtained from (233) as, for p ≥ 1 where we recall that Λ = −e −λ−s . This leads to This formula recovers, as in [31] to the leading order in b, that the typical N (s) at fixed s is large with N (s) typ ∼ 1/b 3/2 . All cumulants are also of the order 1/b 3/2 hence the relative fluctuations are small (but not the absolute ones). As s → −∞, the distribution becomes peaked around N (s) (which is also the asymptotic integrated density of the Airy process) with Gaussian fluctuations with variance N (s) 2 c √ −s πb 3/2 , as the higher cumulants tend to zero (using Li a (z) − (ln(−z)) a Γ(a+1) for z → −∞). The formula (243) thus provides the correction terms in the small b expansion of all cumulants. Note that one can equivalently write where the cumulants κ n are given in (129), with the same substitution (234). One observes that the p-th cumulant of N (s) is only determined by the cumulants κ n with 1 ≤ n ≤ p.
Since we also have an exact expansion to all orders for κ 1 we have, using (110) and (111), or alternatively (112), to all orders in b, where we have used the identity (B3). Let us also give the variance of N (s) to O(b 6 ), either from (243) or from (129) . We note that the terms are obtained by simply shifting L k → −L k+1 in (129) (we recall that σ∂ σ L k (σ) = L k+1 (σ)). Similar manipulations lead to explicit expressions for the expansion of the higher cumulants p ≥ 3. Finally, there is an exact formula for the variance to all orders in b, from (120) we find E. Low temperature: large deviation of the PDF of N (s) in the region s ∼ b 3 We now study the low temperature limit of large b. At strictly zero temperature b = +∞ the typical fluctuations of the position of the rightmost fermion are well known to be given by the Tracy-Widom distribution [15] and some results for the counting statistics near this position have also been obtained [40,45,[48][49][50][51][52][53]. There are however interesting and non-trivial large deviations for large but not strictly infinite b. First of all it is known that (see [35] section 9) for any fixed z < 0 This implies that we expect that the Laplace transform (233) of the PDF of N should take the form, denoting λ = b 3λ where the large deviation rate function Φ(z,λ) is related to F (ν) by the Legendre trans- We now show that there is indeed such a rate function Φ(z,λ) and we calculate it.
As for the calculation of Φ − (z) leading (140) we can retain only the leading order of the cumulants κ 0 n (σ) given in (132). In fact the calculation is very similar to the one of Section III B 3. We must make the replacement (234) and use again the asymptotics (133). We obtain, replacing t = b 3 , where we performed the replacement (234) inside the formula for the leading order of the cumulants (132) and used that L 1 (−e −zt ) t→+∞ − 1 π (−zt) The interpretation of that fact is that whenλ > −z the minimum in the Legendre transform (253) must be attained for ν = 0, a fact that we will check later. We now obtain the rate function Φ(z,λ) forλ < −z as follows. It is convenient to define u 1 = −z > 0 and u 2 = −z −λ > 0. It is written in the form of a series in an expansion in z,λ simultaneously large (equivalently u 1 , u 2 large) as where we have separated the terms n = 1, 2. The only delicate term is n = 2, let us indicate its calculation. It reads (forλ < −z and σ = −e −b 3 z ) Calculation of this integral gives the result for n = 2 in (257). Note that (257) is only a series expansion (to all orders) and we have not obtained Φ(z,λ) in a form as explicit at Let us now extract some information about the PDF P (N ) in (251). The associated rate function is given by the Legendre transform Although we do not have the full explicit form we can use the expansion (257) to obtain F (ν) in an expansion in the limit of large |z|. In that limit ν is large with ν ∼ (−z) 3/2 , hence we defineν through so that for z → −∞ the typical value isν typ = 1. Then one has the expansion at large negative z and fixedν where each term corresponds to a term in the expansion of Φ − (z) at large |z| and F n (0) = 1. Keeping only the leading term in (257) we obtain to leading order F (ν) max We find that F 0 (ν) is convex, with the minimum attained at the typical (i.e. mean) valueν typ = 1, such that F 0 (ν typ ) = 0. The maximum in (262) is achieved at λ =λ * = −z(1 −ν 2/3 ). Note that the partν < 1 corresponds toλ > 0, andν > 1 corresponds toλ < 0. This second part seems also correct: while our calculation adressed λ > 0 its analytical continuation to λ > 0 seems to hold.

VII. CONCLUSIONS
We have studied in this paper short time expansions for large deviations of the height in the KPZ equation on R with either droplet (sharp wedge) or stationary initial condition. In the droplet case, we obtained a systematic algorithm giving the short time expansion. For the Brownian initial condition, our results give the first two orders in time. It would be interesting to see whether the weak noise theory could be extended beyond the leading order, and compare with our results.
Our results for the droplet case extend straightforwardly to more general linear statistics of the Airy point process describing the edge of the spectrum in the Gaussian unitary ensemble. It allows to test the matching to the Gaussian free field in the bulk regime, and to describe corrections due to the edge of the spectrum. Applications to the counting statistics of fermions at the edge of a trap are also discussed, e.g. deviations from the high temperature Gumbel and Poisson distributions are systematically obtained.
Finally, we have pointed out several unexpected similarities hinting at the possible existence of a duality between large deviations around the (perturbative) Edwards-Wilkinson fixed point and the (non-perturbative) KPZ fixed point in a finite volume. A more precise formulation of this hypothetical duality would be extremely nice. It might be helpful for this to understand precisely how the results of [70] for the symmetric simple exclusion process, which have somewhat similar expressions involving half-integer polylogarithms, may fit within the duality.

Sparre Andersen theorem
The Sparre-Andersen theorem lies in the framework of random partial sums S i = X 1 + · · · + X i of a sequence of random variables {X i }. Here we make the hypothesis that our process forms a bridge, i.e. S 0 = S n+1 = 0 and we are interested in N * n be the number of points (j, S j ), j = 1, . . . , n which lie above the straight line from (0, 0) to (n + 1, S n+1 ).
Theorem B.1 (Sparre Andersen Corollary 1, Ref. [71]). If the random variables X 1 , . . . , X n+1 are independent and each has a continuous distribution, or if the random variables are symmetrically dependent and the joint distribution function is absolutely continuous, then for any C which is symmetric with respect to X 1 , . . . , X n+1 and has P(C) > 0, we have ∀m ∈ [0, n], P(N * n = m | C) = 1 n + 1 (B8) For our case of interest, the event C will be our hypothesis that the process is a bridge.
3. Proof of the identities for the functions L β In this section, we prove the identities (50)- (53) for the functions L β (σ, a, b).
a. Proof of the identity (50) From the definition (48), one has where K n (z) = e − z 2 4n √ 4πn is the Gaussian kernel. The integral in the expression above can be interpreted as the probability that a random walker, starting initially at position 0, ends at position 0 after k − 1 steps after staying only on positive positions for all intermediate steps. The transition probabilities depend on the variables n j , but in a symmetric way since each n j has the same weight. Thus, it is possible to use the Sparre Andersen theorem (see Appendix B 2) to replace the integral by 1 k K n 1 +...+n k (0). This leads to L β (σ, 0, 0) = 1 √ 4π ∞ k=1 (−1) k−1 k ∞ n 1 ,...,n k =1 k =1 a n l ,β σ n √ n 1 + . . . + n k With no boundary term using that g(0) = 0 as long as √ xg(σe −x ) → 0 for x → +∞. We precisely obtain (50).
b. Proof of the identity (51) The identity (51) simply follows from renaming z j → z k−j in the definition (48).
c. Proof of the identity (52) The identity (52) is a consequence of (51) and (53). However, since our derivation of (53) below is slightly involved, we prove here (52) directly from the definition (48) of L β (σ, a, b). For any function f , the identity k j=0 ∂ z j k =1 f (z − z −1 ) = 0 holds. After insertion into (48), this implies Performing the integration with respect to z j , exchanging summations as ∞ k=1 k−1 j=1 = ∞ j=1 ∞ k=j+1 and making the change of variable k → k + j leads to (52).

All order expressions
It is possible to obtain an exact expression for κ 3 (g) to all orders. Starting instead from (108) and inserting the form of the propagator given in the last equation in (100), we can perform the summation over p 1 , p 2 , p 3 and a rescaling of k 1 , k 2 , k 3 to obtain One can indeed check that the logarithmic term has the formφ 0 (±y) in the two branches, whereφ 0 (y) is a simple series in y near y = 0. Next we find that y Li 1/2 (e −y 2 ) 2 − π y 2 = ψ(y) =