Point-interacting Brownian motions in the KPZ universality class

We discuss chains of interacting Brownian motions. Their time reversal invariance is broken because of asymmetry in the interaction strength between left and right neighbor. In the limit of a very steep and short range potential one arrives at Brownian motions with oblique reflections. For this model we prove a Bethe ansatz formula for the transition probability and self-duality. In case of half-Poisson initial data, duality is used to arrive at a Fredholm determinant for the generating function of the number of particles to the left of some reference point at any positive time. A formal asymptotics for this determinant establishes the link to the Kardar-Parisi-Zhang universality class.

a Markov jump process. In this contribution we will explore interacting one-dimensional diffusion processes in the KPZ universality class.
As a start we introduce a family of model systems, explain in more detail the conjectures related to the KPZ universality class, and recall the two major results available so far. The main part of our contribution concerns a singular limit, in which the Brownian motions interact only when they are at the same location.
To motivate our model system we start from the potential of a coupled chain, with x = (x 1 , ..., x n ), x j 2 R, and a twice differentiable nearest neighbor potential, V . The precise definition of a point-interaction will be given in Sect. 2, while in the introduction we outline the general picture. To construct a reversible diffusion process with invariant measure the drift is taken to be the gradient of V tot , while the noise is white and independent for each coordinate. Then (1.4) j = 1, ...., n, with the convention that V 0 (x 1 (t) x 0 (t)) = 0 = V 0 (x n+1 (t) x n (t)). Here x j (t) 2 R and {B j (t), j = 1, ...., n} is a collection of independent standard Brownian motions. Note that the measure in (1.3) has infinite mass. Eq. (1.4) is one variant of a Ginzburg-Landau model, see [40] for example.
The dynamics defined by (1.4) is invariant under the shift x j ; x j + a, which will be the origin of slow decay in time. Breaking this shift invariance, for example by adding an external, confining on-site potential V ex as V 0 ex (x j (t))dt in (1.4), would change the picture completely. Just to give one example, one could choose V and V ex to be quadratic.
Then the dynamics governed by Eq. (1.4) is an Ornstein-Uhlenbeck process, which has a unique invariant measure, a spectral gap independent of system size, and exponential space-time mixing. Setting V ex = 0, slow decay is regained. Because of shift invariance, we regard x j (t) as the height at lattice site j at time t. In applications x j (t) could describe a one-dimensional interface which separates two bulk phases of a thin film of a binary liquid mixture. V then models the surface free energy (surface tension) of this interface. If in (1.4) one introduces the stretch r j = x j x j 1 and adopts periodic boundary conditions, then dr j (t) = 1 2 V 0 (r j (t))dt + rdB j (t) , j = 1, ..., n , (1.5) where denotes the lattice Laplacian and r the finite difference operator, both understood with periodic boundary conditions. Clearly, r j (t) is locally conserved and the sum P n j=1 r j (t) is conserved. As a consequence the r(t) process has a one-parameter family of invariant probability measures, indexed by`, which is obtained by conditioning the measure n Y j=1 e V (rj ) dr j , (1.6) on the hyperplane {r | P n j=1 r j = n`}. In the infinite volume limit, the {r j } are i.i.d. with the single site distribution Z 1 e V (rj ) P rj dr j , Z = Z e V (u) P u du , E P (r j ) =`, Point-interacting Brownian motions in the KPZ universality class where E P (·) denotes expectation with respect to the product measure. The parameter P controls the average value of r j . To have Z < 1 for a nonempty interval of values of P , we require the potential V to be bounded from below and to have at least a one-sided bound as V (u) c 1 + c 2 |u|, either for u > 0 or for u < 0, with c 2 > 0. Note that E P (V 0 (r j )) = P , (1.8) which means that P is the equilibrium pressure in the chain. The diffusive limit of (1.5) has been studied in a famous work by Guo, Papanicolaou, and Varadhan [21], who prove that on a large space-time scale the random field {r j (t), j = 1, ..., n} is well approximated by a deterministic nonlinear diffusion equation. The fluctuations relative to the deterministic space-time profile are Gaussian as proved by Chang and Yau [15]. KPZ universality enters the play, when the dynamics (1.4) is modified to become nonreversible. In the physical picture of an interface, the breaking of time reversal invariance results from an imbalance between the two bulk phases which induces a systematic motion. On a more abstract level there are many options. One possibility is to start from a Gaussian process by setting V (u) = u 2 and adding nonlinearities such that shift invariance is maintained and the stationary Gaussian measure of the linear equations remains stationary, see [37] for a worked out example. Here we take a different route by splitting the two drift terms, 1 x j (t)) and 1 2 V 0 (x j (t) x j 1 (t)), not symmetrically but asymmetrically with fraction p to the right and fraction q to the left, p + q = 1, 0  p  1. Then (1.4) turns into (1.9) The totally asymmetric limits correspond to p = 0, 1. One easily checks that for all p the measure (1.3) is still invariant which, of course, is a good reason to break time reversal invariance in this particular way. This property is in analogy to the ASEP, where the Bernoulli measures are invariant independently of the choice of the right hopping rate p. If, as before, one switches to the stretches r j , then dr j (t) = 1 2 p V 0 (r j (t))dt + rdB j (t) , j = 1, ..., n , (1.10) with periodic boundary conditions and 1 2 p f (j) = pf (j + 1) + qf (j 1) f (j). Because of the asymmetry, the macroscopic scale is hyperbolic rather than diffusive. We denote by`(u, t) the macroscopic field for the local stretch r j (t), where u is the continuum limit of the labeling by lattice sites j. Then, using the entropy method of Yau [47], it can be proved that the deterministic limit satisfies the hyperbolic conservation law with P (`) the function inverse to E P (r 0 ) =`. Since`0(P ) < 0, the inverse is well defined.
The limit result leading to (1.11) holds for initial profiles which are slowly varying on the scale of the lattice and up to the first time when a shock is formed. At this point we can explain the striking difference between reversible and nonreversible systems. Let us impose the periodic initial configuration x j =¯j, j 2 Z and, assuming that the dynamics for the infinite system is well defined, let us focus on x 0 (t), the particle starting at the origin. For the symmetric model one expects as t ! 1 with ⇠ G a standard mean zero Gaussian random variable. We are not aware of a completely written out proof, but the key elements can be found in [15]. Harris [22] considers independent Brownian motions, such that the labeling is maintained according to their order. For the dynamics defined by (1.4) this corresponds to the limit of a strongly repulsive potential V with its support shrinking to zero. In [22] it is proved that x 0 (t) is well-defined and that the scaled process ✏ 1/4 x 0 (✏ 1 t) has a limit as ✏ ! 0 which is a Gaussian process with an explicitly computed covariance.
In contrast, for the nonreversible system it is conjectured that in distribution as t ! 1. The anticipated numerical value of c f is explained in Appendix A. Note that, in general, there could be specific values of¯, for which c f = 0. In particular, for the Gaussian process with V (u) = 1 2 u 2 , one obtains P (`) =`and c f = 0 for all`. The random amplitude ⇠ GOE has the distribution function (1.14) Here the determinant is over L 2 (R), P s projects onto the half-line [s, 1), and B 0 is a Hermitean operator with integral kernel B 0 (u, u 0 ) = Ai(u + u 0 ), Ai being the standard Airy function. As proved by Tracy and Widom [42], the expression (1.14) is also the distribution function of the largest eigenvalue of the Gaussian Orthogonal Ensemble (GOE) of real symmetric N ⇥ N random matrices in the limit N ! 1, see [36,19] for the particular representation (1.14).
As in the case of a reversible model, one can regard x 0 (t) as a stochastic process in t. No definite conjectures on its scaling limit are available. We refer to [17] for a discussion.
A proof of (1.13) seems to be difficult with current techniques, except for the Harris limiting case with q = 1. Then the process {x j (t), j 2 Z} is constructed in the following way: for all j, x j (0) = j and x j (t) performs a Brownian motion being reflected at the Brownian particle x j 1 (t). Because of collisions, x 0 (t) is pushed to the right and, as proved in [20], it holds that A second example is the O'Connell-Yor model of a directed polymer in a random medium [32], which has been analysed in considerable detail and again confirms anomalous fluctuations. As before the dynamics is totally asymmetric, q = 1, but the potential is smooth and given by V (u) = e u . Then 16) The initial conditions are x 0 (0) = 0 and, formally, x j (0) = 1 for j 1. As proved in [31], there is a law of large numbers which states that lim t!1 t 1 x butc (t) = (u) a.s. (1.17) for u > 0 with b·c denoting integer part. The limit function can be guessed by realizing that on the macroscopic scale the slope satisfies Eq. (1.11). First note that`= (P ) with = 0 / , the Digamma function. Hence (1.18) see [41] for details. (0) = 0, 00 < 0, and has a single strictly positive maximum before dropping to 1 as u ! 1. Thus t (u/t) reproduces the required singular initial conditions as t ! 0.
Point-interacting Brownian motions in the KPZ universality class Even more remarkable, one has a limit result [6,7] for the fluctuations, lim t!1 The non-universal coefficient (u) will be discussed in Appendix A. Note that the proper rule is to subtract the asymptotic mean value and not the more obvious mean at time t. Hence the limit distribution may have a non-zero mean and in fact E(⇠ GUE ) ⇡ 1.77.
In our contribution we will study interacting diffusions with partial asymmetry and random initial data. As in the previous example, the index j 2 Z + . But we have to resort to point interactions. The precise definition of the dynamics will be given in the following section. As initial conditions we assume that {x 0 , x j+1 x j , j 0} are independent exponentially distributed random variables with mean 1. Hence at t = 0 the macroscopic profile is (u) = u, u 0. For point interactions, one has V = 0 in Eq. (1.7) and thus P (`) =` 1 . The integrated version of Eq. (1.11) reads which for our initial conditions has the self-similar solution with p < 1/2, = q p. Anomalous fluctuations are expected to be seen in the window 0 < u < t not too close to the boundary points.
The three examples discussed above require distinct techniques in their analysis. The first example uses that, upon judiciously choosing dummy variables, there is an embedded signed determinantal process. In the second example one derives a Fredholm determinant for the generating function E exp[ ⇣e xj (t) ] with ⇣ 2 C, <⇣ > 0. In contrast our analysis is based on self-duality of the particle system. x j (t) is replaced by N (u, t), which is the number of particles to the left of u at time t, i.e. the largest j such that x j (t)  u. e is replaced by ⌧ = p/q < 1 and exp by the ⌧ -deformed exponential e ⌧ . Following the strategy in [10], we arrive at a Fredholm determinant for the expectation E e ⌧ (⇣⌧ N (u,t) ) . This is our main result. To establish the connection to KPZ universality, we add a heuristic discussion of a saddle point analysis for this Fredholm determinant.
To prove duality we need some information on the transition probability, which will be provided in a form following from the Bethe ansatz. Such a formula could be of use also in other applications.

Brownian motions with point interactions, self-duality
We consider n interacting Brownian particles governed by the asymmetric dynamics of Eq. (1.9). Point interactions are realized through a sequence of potentials, V ✏ , which are repulsive, diverge sufficiently rapidly as |u| ! 0, and whose range shrinks to zero as ✏ ! 0. More precisely, we start from a reference potential (u)  0 for u > 0, and, for some > 0, lim u!0 |u| V (u) > 0. The scaled potential is defined by V ✏ (u) = V (u/✏) and the corresponding diffusion process is denoted by y ✏ (t). Since the potential is entrance -no exit [30], the positions can be ordered as y ✏ 1 (t)  ...  y ✏ m (t). Hence y ✏ (t) 2 W + m , the Weyl chamber in R m such that the left-right order is according to increasing index. Since the particle order is preserved, we deviate slightly from the viewpoint of the introduction and regard the positions of particles as a point configuration in R. As will be proved in Appendix B, there exists a limit process, y(t) 2 W + m , such that lim ✏!0 y ✏ (t) = y(t).
is the right-sided local time accumulated at the origin by the nonnegative martingale y j+1 (·) y j (·). So y j (t) is pushed to the left with fraction p of the local time whenever y j (t) = y j+1 (t) and it is pushed to the right with fraction q of the local time whenever y j (t) = y j 1 (t), which implies that the drift always pushes towards the interior of W + m .
If q = 1, y j+1 (t) is reflected at y j (t). In particular, y 1 (t) is Brownian motion. If q = 1/2, the dynamics corresponds to independent Brownian motions with ordering of labels maintained. In [28] it is proved that (2.1) has a unique strong solution. Furthermore, triple collisions, i.e. the sets {y j (t) = y j+1 (t) = y j+2 (t) for some t}, have probability 0.
with E y denoting expectation of the y(t) process of (2.1) starting at y 2 W + m . As proved in Section 6, it holds the directional derivative being taken from the interior of W + m . q = 1/2 corresponds to normal reflection at @W + m . With this boundary condition y is a self-adjoint operator. q 6 = 1/2 is also referred to as oblique reflection at @W + m [45,23].
In addition to the y-particles we introduce n dual particles denoted by (x 1 (t), ..., x n (t)) = x(t). They are ordered as x n  ...  x 1 , hence x 2 W n , the Weyl chamber in R n such that the left-right order is according to decreasing index. For the dual particles the role of q and p is interchanged. Thus their dynamics is still governed by (2.1) with ⇤ (j,j+1) (·) = L xj xj+1 (·, 0). Also the boundary condition (2.5) remains valid, the directional derivative being taken from the interior of W n .
The main goal of this section is to establish that the x(t) process is dual to the y(t) process. The duality function is defined by (2.6) where ⌧ = p/q and throughout we restrict to the case 0 < ⌧ < 1. ✓(u) = 0 for u  0 and ✓(u) = 1 for u > 0. Such type of duality is known also for other stochastic particle systems [26], in particular for the ASEP [10].
Proof. We first compute the distributional derivative of H. Setting @ x↵ = @/@x By interchanging x ↵ and y one arrives at Correspondingly for the derivative w.r.t. y , , the set of all twice continuously differentiable functions vanishing rapidly at infinity and with boundary conditions As will be discussed in Section 6, the generator L x of the diffusion process x(t) is given by L x = 1 2 x on the domain D(L x ) and correspondingly for L y . The integral kernel of e Lxt , denoted by P x (dx 0 , t), is the transition probability for x(t). It has a density, P .., n 2} ⇥2 . Correspondingly P + y (y 0 , t) defines the transition density for the y(t) process. (2.12) By the fundamental theorem of calculus, for 0 < ✏ < t ✏, (2.14) Point-interacting Brownian motions in the KPZ universality class By Lemma 2.2 and for ✏  s  t ✏ the function and correspondingly for y. Hence one can differentiate in (2.14) and obtains Since the transition probabilities are smooth, L x H and L y H can be obtained as distributional derivatives. Hence We integrate Eq. (2.18) against the smooth function f 1 (x)f 2 (y). By continuity we can take the limit ✏ ! 0. The integrand of the resulting identity is continuous in x, y and the identity (2.7) holds pointwise.
Remark. An alternative proof, based on ASEP duality, is discussed in Appendix C.

Half-line Poisson as initial conditions, contour integrations
We want to study the y(t)-process in case the particles are initially distributed according to a Poisson point process with density profile ⇢(u) = ✓(u). By space-time scaling, the density 1 on the half-line could be changed to any other value. The initial macroscopic height profile is then h(x, 0) for x  0 and h(x, 0) = x for x > 0. In the course of time the wedge is expected to smoothen with superimposed KPZ fluctuations characteristic for droplet growth. Since for such initial condition the y(t)-process has an infinite number of particles, our previous results cannot be used directly. So let us choose the density of the initial Poisson point process as ⇢`(u) = 1 for 0  u à nd ⇢`(u) = 0 otherwise, which results in a finite number of particles. Let us denote by N (u; y) the number of particles in the configuration y located in ( 1, u] and set N (u, t) = N (u; y(t)) as a random variable. We first average the duality function over the Poisson point process with density ⇢`, Here E`refers to the y(t)-particle process with initial Poisson of density ⇢`.
Since N (u, t) 0 and ⌧ < 1, the moments on the left hand side of (3.3) determine uniquely the distribution of N (u, t). Let us denote the corresponding random variable by N`(u, t). For fixed n in the limit`! 1,Fǹ converges toF n ,Fǹ(t) converges toF n (t), and the expression on the right hand side of (3.3) converges toF Next we provide a formula forF n (t) at general arguments.
Remark. It is understood throughout that the contour integration includes the We consider the boundary condition (2.13) with directional derivative taken from (W n ) . One has The integrand has no poles in the strip bordered by C`and C`+ 1 . Hence C`can be moved on top of C`+ 1 . The integrand is odd under interchanging z`and z`+ 1 and the right hand side of (3.7) vanishes.
From the explicit form on the right hand side of (3.5) we infer that F n is bounded and continuous.
(ii) initial conditions. We have to show that lim . Note that the integrand in (3.5) has an integrable bound at infinity uniformly in t and hence one can set t = 0. We define the sector S`by with`= 1, ..., n. ThenF and F n (x, 0) will be computed for the sector S`. Since 0 < x`< ... < x 1 , exp(x j z j ) decays exponentially as <z j ! 1, j = 1, ...,`, and the contours C 1 , ..., C`can be deformed to circles around z = (1 ⌧ ), maintaining the nesting condition. Correspondingly, since x n < ... < x`+ 1 < 0, the contours C`+ 1 , ..., C n can be deformed to circles around z = 0, maintaining the nesting condition.
We integrate first over z 1 . Then on S`, denoting the deformed contours byC Iterating the integrations over z 2 , ..., z`yields Next we integrate successively over z n up to z`+ 1 . Abbreviating we adopt an argument of Warren in a similar context [46]. Let us consider F n (x(t), 14) The dt term vanishes because of (3.6) and the Skorokhod term vanishes, because F n satisfies the boundary condition (2.13). Hence with F n (t) as defined in (3.5).

From moments to a Fredholm determinant
To conform with the notation in [7] we relabel z 1 , ..., z n to z n , ..., z 1 . Then, by the results of Sect. 3, The goal of this section is to obtain a Fredholm determinant for the ⌧ -deformed generating function of ⇣⌧ N1(u,t) , i.e., The required definitions for ⌧ -deformed objects are well summarized in Appendix A of [10].
contour j contour j Figure 1: A single move in unnesting the contours. Displayed is only the move of contour j across the singularity at the gray dot ⌧ z j+1 generated by a fixed point on contour j + 1.
The first step is to remove the nesting constraint by moving the contours. In Fig. 1 we display a single move. z j+1 is fixed and the integration is over z j . The singularity for z j is at ⌧ z j+1 . We deform the z j contour across the singularity and thereby pick up a pole contribution, which is evaluated by the residue theorem. The resulting combinatorial structure is identical to that of Proposition 3.2.1 in [6]. In the case of unbounded contours, the same combinatorial identity is stated in Proposition 4.11 of [7] upon identifying µ n with E ⌧ nN1(u,t) . The function f (z) in Proposition 4.11 is our f (z; u, t), which has a single pole at (1 ⌧ ) implying the simplification N = 1. Hence The w j -contours are all the same and given by C r = { + i', ' 2 R} with 0 < < 1 ⌧ . The notation `n above means that partitions n, i.e. if = ( 1 , 2 , ...) then n = P i , Point-interacting Brownian motions in the KPZ universality class and the notation = 1 m1 2 m2 ... means that i shows up m i times in the partition . Not unexpected, for the ASEP a similar type of formula holds [10,9] Rearranging terms and using the ⌧ -binomial theorem, one arrives at a Fredholm determinant of the ⌧ -deformed generating function (4.3).
where the operator K is defined through the integral kernel Proof. In its algebraic steps the proof is identical to the one given in Section 7.1 of [7].
We only need to verify that the Fredholm expansion is well-defined also in our case.
Clearly, for all n 1 and on C r , for constants c 2 , c 3 > 0. This ensures the convergence of the Fredholm expanded determinant for sufficiently small |⇣|.
The Fredholm determinant in (4.5) is not yet suitable for asymptotics and one has to replace the sum over n by a contour integral, which is achieved by a Mellin-Barnes type integral representation, see Lemma 3.2.13 of [6]. We introduce h as the solution of f (z, t) = h(z, t)/h(⌧ z, t). Then We also need the integration contour, C w , as displayed in Fig. 2. This contour is reflection symmetric relative to the real axis and piecewise linear with starting point 1 2 , moving then to 1 2 + id, then to R + id, and finally to R + i1, d > 0, R 1 2 .
with w, w 0 2 C r and C w as in Lemma 4.2. Then Proof. Our theorem is in close analogy to Theorem 4.13 in [7]. When comparing Eq.
(4.13) of [7] for the particular case N = 1 with our Eq. (4.9), one notes that, upon setting q = ⌧ , the function g the exponential factor, for which exp w(q s 1) is to be replaced by exp u(1 ⌧ s )w + 1 2 (4.14) The s-integration is along the same contour. Only the w-integration is along e C↵ ,' in [7], while we integrate along C r . For the proof of Theorem 4.13 the properties of the exponential factor are used only in Eq. (7.6) of [7], which is replaced by The ' 2 term thus dominates the linear term. Our Gaussian bound replaces the exponential bound of Eq. (7.6) in [7]. The remainder of the proof follows verbatim Section 7.2 of [7].

Formal asymptotics
To obtain the long time asymptotics of N 1 (u, t/ ) requires a steepest decent analysis of the kernel K ⇣ of (4.12). Here we only identify the saddle point and its expansion close to the saddle. Thereby the GUE asymptotics becomes visible. For a complete proof a more detailed analysis of the steepest decent path would have to be carried out. There are other models in the KPZ class for which such kind of analysis has been accomplished, see [1,7,20,8] as examples.
One first has to figure out the law of large numbers for N 1 (u, t). The quick approach is to use the ASEP, 1 2 < q  1, with step initial conditions. On the macroscopic scale the density, ⇢, is governed by @ t ⇢ @ u (⇢ ⇢ 2 ) = 0 , see [4]. In the low density limit one has to shift to the moving frame, which amounts to substituting ⇢ by⇢(u, t) = ⇢(u + t, t). Then⇢ satisfies The solution with initial data⇢(u, 0) = ✓(u) reads⇢(u, t/ ) = u/2t for 0  u  2t. We scale u = at with a > 0 and eventually t ! 1. Then to leading order N 1 (at, t/ ) = 1 4 a 2 t , 0  a  2 , N(at, t/ ) = (a 1)t , 2  a . For a > 2 one expects to have Gaussian fluctuations of size p t, while for a < 2 the fluctuations should be KPZ like of size t 1/3 . In the following we restrict to 0 < a < 2. The same law of large numbers can be obtained from Eq. (1.11) for the macroscopic stretch , by noting that P (`) =` 1 for point interactions.
We substitute z = ⌧ s w, s log ⌧ = log z log w and set Inserting on the left hand side of (4.13), it follows, see [7], Lemma 4.1.39, that Thus we have to study the corresponding limit on the right hand side of (4.12). In the new coordinates the kernel reads where G(z) = 1 2 z 2 az 1 4 a 2 log z .

Point-interacting Brownian motions in the KPZ universality class
There is an extra factor (z c t 1/3 ) 1 from the volume element due to the change in w, w 0 . We substitutez,w,w 0 by (a/2) 2/3 z, (a/2) 2/3 w, (a/2) 2/3 w 0 and thereby arrive at the limiting kernel The w contour is now given by two rays departing at 1 at angles ±⇡/3, oriented with increasing imaginary part, and the z contour is given by two infinite rays starting at 0 at angles ±2⇡/3, oriented with decreasing imaginary part. The Fredholm determinant with this kernel is identical to the Fredholm determinant of the Airy kernel, see [44] Lemma 8.6. Hence one concludes that with F GUE (r) = P(⇠ GUE  r), under the assumption that the contribution from the remainder of the steepest decent path vanishes as t ! 1.

The Bethe ansatz transition probability
The goal of this section is to establish that the dynamics with point interactions has a "smooth" transition probability, as used in Section 2 for the proof of duality. While there should be a more abstract approach, we will use the Bethe ansatz construction of the transition probability, as pioneered by Tracy and Widom [43,44] in the context of the ASEP. To make the comparison transparent, we follow closely their notation, which in part deviates from earlier notations. The particle process is denoted by x(t) 2 W + N with initial condition x(0) = y. As explained before x(t) is the semi-martingale determined by is the right-sided local time accumulated at the origin by the nonnegative martingale with E y denoting expectation of the x(t) process of (6.1) starting at y 2 W + N . As to be shown, f satisfies the backwards equation the directional derivative being taken from the interior of W + N . Let us define the standard decomposition P x(t) 2 dx x(0) = y = P y (x, t)dx + P sing y (dx, t) . In spirit P y (x, t) should be the solution to the backwards equation. We follow Bethe [5] and start from an ansatz for the solution of (6.4), (6.5) given by where the sum is over all permutations of order N . The Gaussian factor ensures that Eq. (6.4) is satisfied. The expansion coefficients A are determined through the boundary condition (6.5). We define the ratio of scattering amplitudes for wave numbers z ↵ , z 2 C. The expansion coefficient A can be written as  For 1 < ⌧ < 1, Eq. (6.10) still holds, but one has to impose a < 0. The limiting cases ⌧ = 1 and ⌧ ! 0 will be discussed below.
In the remainder of this section we prove Theorem 6.1 with some estimates being shifted to Appendix D.
We first investigate properties of Q y (x, t) and set with properties of the test function f to be specified later on.     The validity of Eq. (6.12) is easily checked. For the boundary condition we note upon interchanging z 1 and z 2 . Clearly, I 12 (y; x, t) satisfies (6.14). Thus we still have show that lim t!0 I 21 (y; x, t) vanishes for y 2 W + 2 and x 2 (W + 2 ) . For this purpose, we introduce a new variable, z 0 , by z 0 = z 1 + z 2 and substitute z 2 by z 0 . Then The pole of z 1 is at z 1 = (1 + ⌧ ) 1 z 0 and hence to the right of a . Under our assumptions one has x 2 x 1 + y 2 y 1 > 0.
Proof of Lemma 6.2: Property (6.12) is easily checked. The argument leading to (6.13) is identical to Theorem 2.1, proof of (b), in [43]. Property (6.14) follows directly from the definition. The difficult part is (6.15). In fact, for the ASEP the analogue of I is not necessarily equal to 0 and one has to use cancellations. In this respect the contour integral for Brownian motions with oblique reflections has a somewhat simpler pole structure than its lattice gas version. We where ⇢ ⇢ z`means that this term is omitted from the sum. Since ⌧ < 1, the pole for the zì ntegration lies to the right of a . Furthermore, if j 6 =`, the denominator reads z` As before, the pole for the z`integration lies to the right of a . In the second case a generic factor reads with ↵ > . For the z`-integration either`= ↵ or`= need to be considered. If`= ↵, then`> . Since`= min B, one must have 2 A. But then (`, ) is not an inversion. Hence`= and the pole for the z`integration is at ⌧ 1 z ↵ for some ↵ 2 [1, ..., N 1] and hence to the right of a . Thus the z`integration has no poles to the left of a . If`= , then ↵ >`and the argument just given applies. With this information Property (6.15) can be proved, but we leave the details for Appendix D.
Proof. Let us denote P y (f, t) = E y f (x(t)) . We have to show that P y (f, t) = Q y (f, t), which corresponds to Theorem 3.1 upon identifying P y (f, t) with F n and Q y (f, t) withF n .
We have established already that Q y (f, t) satisfies the properties (i) and (ii) in the proof of Theorem 3.1. In addition x 7 ! Q x (f, ⌧ ) is continuous and bounded. So we merely have to copy part (iii) with the result Continuously in ✏, x(✏) ! y and P x(T ) (f, ✏) ! f (x(T )). Hence Q y (f, T ) = E y f (x(T )) .  Proof. In (6.29) we insert the decomposition in (6.6). Then (6.33) which implies that P y (x, t) = Q y (x, t) a.s. and P sing y ((W + N ) , t) = 0. To prove that the singular part vanishes, by normalization one only has to establish that Q y (1 1, t) = 1 with 1 1(x) = 1. We set g g N (u) is the distribution function for the N -th particle at fixed initial configuration y. Since a > 0, all x-integrals are convergent and Following [43], one can rewrite To apply the first combinatorial identity of Tracy and Widom [43], Section VI, one has to invert the order as˜ (j) = (N j). Then (6.35) reads In the second line we used the combinatorial identity in the limit ⇠ j = 1 + z j to linear order in z j . Inserting in (6.35), one arrives at We have to show that lim u!1 g N (u) = 1. We integrate over z 1 . The poles for z 1 are at ⌧ 1 z j , z j 2 a , j = 2, ..., N, and at z 1 = 0. We choose u sufficiently large such that u y j > 0. Then the contour a can be deformed to a contour˜ a plus a small positively oriented circle around 0.˜ a coincides with a far away from the origin and lies to the left of z 1 = 0 close to the origin. Integrating along the circle yields g N 1 (u) and one arrives at the identity (6.39) In the limit u ! 1 the first summand vanishes, since all poles of the z 1 -integration are to the right of˜ a . Hence lim u!1 g N (u) = lim u!1 g N 1 (u). But lim u!1 g 1 (u) = 1 and the claim follows by induction.
This concludes the proof of Theorem 6.1.
There are two limiting cases of interest, ⌧ ! 1 which corresponds to the symmetric interaction and ⌧ ! 0 which corresponds to the maximally asymmetric interaction. In the limit ⌧ ! 1 one has S(z ↵ , z ) = 1. with the Gaussian kernel p t (u) = (2⇡t) 1/2 exp( u 2 /2t) and perm denoting the permanent, i.e. omitting the factor sgn in the definition of the determinant.

Point-interacting Brownian motions in the KPZ universality class
The contribution of Harris [22] relies on the formula (6.40). The limit ⌧ ! 0 of the transition probability has been first written down in [39], see also [46]. Corollary 6.6. For q = 1 where for m 2 Z Proof. For q = 1 the integrand in (6.7) reads results in (6.41).

A Non-universal constants
The asymptotics in (1.13) is the sum of two terms. The deterministic term is proportional to t. Its prefactor can be guessed on the basis of the Hamilton-Jacobi equation for the height, = q p, compare with (1.11). The solution to (A.1) should be of the self-similar form, h(x, t) = t (x/t), for large t. Then the reference point is chosen as x = ut and to leading order the height grows linearly in t. Such structure can be achieved for wedge initial conditions including the degenerate linear profile, h(x, 0) =`x, which is referred to as either flat or stationary initial condition. The fluctuating part of (1.13) is more difficult. Here our conjecture relies on a particular model with exact solutions. The respective formula can be put in a form which makes its generalization evident and can be checked against a few other models. In fact, the conjectures are really based on the universality hypothesis for models in the KPZ class. In our context the hypothesis states that, for 6 = 0, the fluctuation properties are independent of the choice of the interaction potential V , except for potential dependent scales. The non-universal prefactors listed below could possibly vanish, in which case a more detailed analysis is required.
We discuss separately the three canonical cases, wedge, flat, and stationary initial conditions.
(i) wedge initial conditions. We consider two initial wedges, labelled by = +, and given by with` <`+ and denote by h (x, t) the corresponding solution of (A.1). Our initial value problem is equivalent to the Riemann problem for a scalar conservation law in one dimension, which is well studied, see [24], Chapter 2.2, for a detailed discussion. is linear outside the interval [y , y + ] with slope` to the left and` to the right of the interval. Inside the interval there are finitely many cusp points, i.e shocks for the slope. We label them as y < y 1 < ... < y k < y + , where the cases y < y + , no cusp point, and y = y + are admitted. Then h (x, t) is

self-similar and reads
h (x, t) = t (x/t) .
⇠ GUE has a negative mean and the actual interface is more likely located towards the interior of tangent circle at (u, (u)). If P 00 has a definite sign, then one of the two cases is empty. But in general either case has to be considered.
Our conjecture is based on the KPZ equation, from which the non-universal coefficients follow immediately by its scale invariance [1,38]. The result has been confirmed by the TASEP with step initial conditions [27] and a variety of similar models [7,41]. Since there is no exact solution for the KPZ equation available, this time we use as reference model the TASEP with a periodic particle configuration as initial condition [36,12]. The resulting formula has been checked for a few other models [11,20]. The Baik-Rains distribution function, F BR (s), also denoted by F 0 (s), is defined in [2,33]. As far as known, it is not related to any of the standard matrix ensembles.
In (A.7) and (A.8) the reference point butc is arbitrary, while (A.9) only close to the characteristic of Eq. (A.1) one observes the anomalous t 1/3 scaling. Away from the characteristic the fluctuations would be Gaussian generically.
The asymptotics of the KPZ equation with stationary initial data has been accomplished recently [8]. By scaling the result (A.9) follows, which is then confirmed through the TASEP [2,34,19] and the stationary version of the model defined in (1.16) [8].

B Convergence to point-interaction
The main text concerns Brownian motions with oblique reflection. In the introduction we argued that such point-interaction can be approximated through a short range, sufficiently repulsive potential. Here we prove such a claim. To keep matters simple, we only establish convergence of the second moments.
with some constant c 0 independent of ✏. Thus E( ✏ j (t)) is bounded uniformly in ✏.

C Low density ASEP
We explain an alternative proof of Theorem 2.1 based on ASEP duality.

(C.3)
(i) The approximation theorem. It suffices to discuss the particle process y(t). We consider m ASEP particles with positions w 1 (t) < ... < w m (t), w j (t) 2 Z. Particles jump with rate p to the right and rate q to the left, subject to the exclusion rule. Switching to the moving frame of reference and under diffusive rescaling one obtains with b·cdenoting integer part. Clearly y ✏ j (t) 2 (W + m ) \ (✏Z) m . Proposition C.2. Let f : W + m ! R be bounded and continuous. Then for initial conditions y ✏ such that y ✏ ! y 2 W + m it holds lim ✏!0 E y ✏ f (y ✏ (t)) = E y f (y(t)) .

(C.5)
In [28], the proposition is proved for the asymmetric zero range process with constant rate, c(n) = 1 0n , which differs from the ASEP at most by m uniformly in t. (ii) ASEP duality. We introduce n dual particles. They jump with rate q to the right and rate p to the left, subject to the exclusion rule. The diffusively rescaled positions of the dual particles in the moving frame are denoted by x ✏ j (t). Proposition C.3. For all x 2 (W + m ) \ (✏Z) m and y 2 (W + m ) \ (✏Z) m it holds E ✏ x H(x ✏ (t), y) = E ✏ y H(x, y ✏ (t)) .

(C.6)
In [10] the assertion is proved for ✏ = 1 at fixed lattice frame. In (C.6) the y ✏ (t) frame moves with velocity p q, while the x ✏ (t) frame with velocity q p. To check that the terms just balance one uses that ✓( u) = ✓(u) for > 0 and the translation invariance of the ASEP dynamics.

D Proof of (6.15)
We fix , 6 = id, n, hence the sets A, B, and`= min B. We have argued already that the integration over z`results in an expression vanishing as t ! 0. To have a proof we have to study the full 2N -dimensional integral. For f 2 D ✏ , this integral reads The strategy is to first integrate over w _`w hich results in g(w, z _`) , where by construction g is supported in [✏, 1) in dependence on w and is smooth with a rapid decay on the contours a , aN . Secondly we bound the integration in dz`dw with an explicit dependence on z _`. For this purpose we have to study the S-factors. One has (D.14) Since |✓( w)e bj w | < 1, one obtains the bound of (D.10).F j is supported on ( 1, 0], f on [✏, 1), and lim t!0 p t (w) = (w), which establishes the limit of (D.10).