On the average of the Airy process and its time reversal

We show that the supremum of the average of the Airy process and its time reversal minus a parabola is distributed as the maximum of two independent GUE Tracy-Widom random variables. The proof is obtained by considering a directed last passage percolation model with a rotational symmetry in two different ways. We also review other known identities between the Airy process and the Tracy-Widom distributions.

also explicit and is given by a determinantal formula involving the Airy function. The Airy process is stationary but is not Markovian.
In addition to the above basic connection, there are interesting identities between the supremum of a function of the Airy process and the Tracy-Widom distribution functions. The purpose of this paper is to establish one more such an identity. We first review two known identities. Theorem 1.1 ( [19]). For every x ∈ R, (2) Theorem 1.2 ( [5]). Let A (1) and A (2) be two independent Airy processes. Then for every α, β > 0 and for every x ∈ R.
The main result of this paper is the following identity.
Hence the supremum of the average of the Airy process and its time reversal minus a parabola is distributed as the maximum of two independent GUE Tracy-Widom random variables. A direct proof of Theorem 1.1 was recently established in [13]. The authors of [13] first extended the determinantal formula for the joint distribution of A(τ ) at finitely many times to a determinantal formula for P(A(τ ) ≤ g(τ ), t ∈ [−T, T ]) for general function g and T > 0, and then showed how Theorem 1.1 can be obtained from this formula when g(τ ) = τ 2 + x. It is an interesting open question to find a similar direct proof for Theorem 1.2 and Theorem 1.3 as well as Theorem 1.5 and Theorem 1.6 below. The reader is referred to [23] for a survey of this direct proof and also other identities for the cousins of the Airy process.
There is one more known identity for the Airy process. 24]). For every w ∈ R and x ∈ R, where G 2→1 w (x) is the marginal distribution function of the process A 2→1 introduced in [10] (see (1.7) of [24]).

The distribution G 2→1
w (x) interpolates F 2 and F 1 : It converges to F 2 (x) as w → −∞ and to F 1 (4 1/3 x) as w → +∞. The paper [24] gave a direct proof of (5) using the method of [13]. In terms of DLPP models, this identity can be obtained by considering a point-to-half line last passage time by using the result of [10].
In all of the above identities, the distribution of the argmax, τ max , at which the supremum is attained is also of great interest since it describes the transversal fluctuations of the associated DLPP model. The distribution of τ max for the identity (2) was computed in two recent papers [20] and [26] independently. The paper [20] is mathematical and rigorous while [26] is physical. The density functions obtained in these papers, which look very different, are subsequently shown to be the same in [4]. It is an interesting open question to find the distribution of argmax for other identities.

Airy process plus Brownian motions
The indirect method for Theorem 1.1, 1.2 and 1.3 can also be used to prove other identities involving the Airy process and the Brownian motion if we consider DLPP models with special rows and columns. We mention two known results. Definition 1.3. For real parameters w + and w − , let F st (x; w + , w − ) denote the distribution function defined as H(x; w + , w − ) in Theorem 3.3 and Definition 3 of [6].
An explicit formula is included in Section 4 below. The function F st (x; w + , w − ) is symmetric in the parameters w + and w − . This function is the limiting distribution for the fluctuations of the height of totally exclusion processes starting with Bernoulli initial conditions (see, e.g., [21]). The parameters w + and w − are associated to the initial density of the particles on the positive and negative parts, respectively. Especially F st (x; w, −w) appears when the initial condition is random and stationary. The following identity is obtained recently. Theorem 1.5 ( [12]). Let B(τ ), τ ∈ R, be a two-sided standard Brownian motion with B(0) = 0. Then for every x ∈ R and w + , w − ∈ R.
These functions are invariant under the permutations of the parameters w 1 , · · · , w k . They are the limiting distributions of the fluctuations of the largest eigenvalue of the so-called spiked random matrix models. It is known that [6] and (1.21) of [1]) and F spiked Theorem 1.6 ( [12]). Let B 1 (τ ), B 2 (τ ), · · · , τ ≥ 0, be independent standard Brownian motions. Let w 1 , w 2 , · · · be real parameters and letB i (τ ; the Brownian motion with drift. Then for every x ∈ R and k = 1, 2, · · · , P sup 2 Proof of Theorem 1.3 As mentioned above, this proof is similar to that of [19] for Theorem 1.1. Fix a parameter q ∈ (0, 1). We use the notation X ∼ Geom(q) to indicated that X is a (shifted) geometric random variable which has the probability mass function (1 − q)q k , k = 0, 1, 2, · · · .
In [8], five types of symmetries of DLPP models with geometrically distributed weights were considered and the limit law for the fluctuations of the last passage time was obtained for each case. One of the symmetry types is the rotational symmetry described as follows. Let R N := {(i, j) : i, j = 1, · · · , 2N }. To each (i, j) ∈ R N we associate a weight w(i, j) with the condition that Apart from the above symmetry conditions, we assume that the random variables are independent and are distributed as Geom(q), that is, N } are independent and distributed as Geom(q), and w(i, j) for (i, j) ∈ R N \ H N are defined by the symmetry condition (10). Let G · (N ) denote the last passage time from (1, 1) to (2N, 2N ) in this model: where π is an up/right path from (1, 1) to (2N, 2N ) consisting of sites whose coordinates increase weakly and W (π) is the weight of path π. where The Poisson version of this percolation model is related to the combinatorial problem of finding the longest increasing subsequence of random signed permutations and also the so-called 2-colored permutation (see, e.g. Remark 2 in Section 1 of [8]), and the limit theorem analogous to the above was obtained in [29,9]. We now consider G · (N ) in a different way. Let } be the sites on the straight line joining (1, 2N ) and (2N, 1). Every up/right path from (1, 1) to (2N, 2N ) intersects L N at a unique point. If we consider the paths which intersect L N at a specific point (i, 2N +1−i), then the maximal weight among these paths equals the sum of two point-to-point last passage times, one from (1, 1) to (i, 2N +1−i) and the other from (i, 2N +1−i) to (2N, 2N ), minus the weight at (i, 2N + 1 − i), which is counted twice. But due to the symmetry conditions (10), the last passage time from (i, 2N + 1 − i) to (2N, 2N ) equals the last passage time from (1, 1) to (2N + 1 − i, i). Therefore, considering all possible intersection points, we find that G · (N ) has the same distribution as where G(m, n) is the usual point-to-point last passage time from (1, 1) to (m, n) with i.i.d. (shifted) geometric weights with no symmetry conditions and |D N | is bounded by the maximum of N independent Geom(q) random variables, which is due to the double-counting of the weights on L N . Since D N = O((log N ) 1+ǫ ) with probability 1 as N tends to infinity for any ǫ > 0, we obtain, inserting (14) into (12), that Since G(m, n + 1) and G(m + 1, n) are between G(m, n) and G(m + 1, n + 1), we can change N + 1 to N and find We show that the left-hand side equals the left-hand side of (4). The limit of G(m, n) is well known. It was shown in [17] that in expectation and also in probability as m, n → ∞ such that m/n is in a compact subset of (0, ∞). This implies, after a simple calculation, that the maximum in (14) is attained at u near 0. This statement can be made precise: see (21) below. Now if we scale u = N 2/3 τ and consider G(N + N 2/3 τ, N − N 2/3 τ ) as a process in "time" τ , then its fluctuations converge to the time-scaled Airy process minus a parabola in the finite distribution sense as well as in the functional limit sense [19]. (Here N 2/3 is the scale of the so-called transversal fluctuations of KPZ universality class [18,3].) More precisely, if we set 2 It remains to show that the left-hand side equals the left-hand side of (4). The functional limit theorem mentioned above implies that for any fixed T > 0. The tail estimate of H N (τ ) for large |τ | was also obtained in [19]. The analysis in that paper implies that (see (127) in [5] for details) for every fixed x ∈ R and ǫ > 0, there are T 0 and N 0 such that for all T > T 0 and N > N 0 . Since P max |τ |≤T (20) and (21) imply that the left-hand side of (19) equals the left-hand side of (4). Theorem 1.3 is proved.
Remark 2.1. In addition to the symmetry · , four other symmetry types were considered in [8]. They are indicated by symbols , , , and . The first one has no symmetry. If we consider the last passage time in this case as above, we arrive at Theorem 1.2: the maximal path in T N := {(i, j) : i + j ≤ N + 1} and R N \ T N give rise to two independent Airy processes since the last passage times in two parts are independent to each other. The second symmetry type has the reflection symmetry about the line L N . Since the maximal path in T N and R N \ T N are identical except for the asymptotically negligible contribution from the site on L N , their sum is basically twice of the maximal path in T N , thus giving rise to single Airy process. This leads to Theorem 1.1.
Remark 2.2. The symmetry types , and contain an extra reflection symmetry about the other diagonal line {(i, j) : j = i}. Hence we can consider these cases as directed last passage models in the triangle T := {(i, j) : j ≤ i}. The limiting distribution for and is F 4 and F 2 , respectively. Now the analog of (18) in triangle T does not converge to Airy process but to a different process 2 In (1.8) of [19], σ is given as . This is a typographical error. The correct formula of σ is as in [17].
which interpolate F 4 and F 2 [25]. Thus we can expect that the the supremum of two independent such processes minus a parabola, over τ ∈ [0, ∞), equals F 4 , and the supremum of one such process minus parabola equals F 2 . To make this statement rigorous, we need the functional theorem and the tail estimate for DLPP in triangle T . This will be discussed in future work. We note that by making the weights on the diagonal line {(i, j) : j = i} different from the rest of the triangle, we can obtain identities for yet another process.
3 Outline of proof of Theorem 1.5 and 1.6 In this section, we outline the proof of Theorem 1.5 and 1.6 obtained in [12] in order to illustrate how different DLPP models give rise to different identities. The basic idea is same as the previous section and [19]: we consider a DLPP model for which the limit theorem is proved, and then interpret the last passage time as the maximum of last passage times of paths with arbitrary endpoints. The technical part is the functional limit and the tail estimate. It turned out that for Theorem 1.5 and 1.6 one needs the functional limit theorem and the tail estimates along horizontal lines and vertical lines instead of the diagonal line in the previous section. Such results are obtained in [12] by proving the so-called slow decorrelation phenomenon and using the results of Johansson [19]. For Theorem 1.5, one uses the following DLPP model: the weights are independent and satisfy where q ∈ (0, 1) is a fixed parameter and for fixed real parameters w + and w − . Here σ is same as (13). It was shown in Section 4 of [6] that the last passage time X(N ) from (0, 0) to (N, N ) satisfies We now consider the last passage time in a different way. The last passage path, considered as starting at (N, N ) and ending at (0, 0), either arrives at a point (i, 0) for some i and travel horizontally left to (0, 0), or arrives at a point (0, j) for some j and travel vertically downward to (0, 0). Hence, denoting by G(N − i, N − j) the last passage time from (N, N ) to (i, j), we find that where S + i is the sum of i independent Geom(α + √ q) random variables and S − j is the sum of j independent Geom(α − √ q) random variables. From (17) and the law of large numbers of independent variables, it is reasonable to expect that the maximum of the above expression occurs when i and j are close to 0. Now set where d is defined in (18). ThenH N (τ ) converges toÂ(−τ ) in the sense of finite distribution and also in the functional limit sense [12]. On the other hand, since the mean and the variance of Geom(a) is a 1−a and a (1−a) 2 respectively, we find from the central limit theorem, after inserting (23), that in distribution where B + (τ ) and B − (τ ) are independent standard Brownian motions. Thus, we find that, at least formally, X(N )−µN This argument was made rigorous in [12]. After we change τ to −τ , this, combined with (24), implies Theorem 1.5.

Formula of distribution functions
For the convenience of the reader, we include the formulas of the Tracy-Widom distribution functions and also the distribution functions in Definition 1.3 and 1.4. All of them have at least two different expressions. One expression involves the Painlevé equation and another expression involves a Fredholm determinant of an operator whose kernel is related to the Airy function. Here we only present the formulas involving the Painlevé equation.
Let q(x) be the solution to the Painlevé II equation q ′′ = 2q 3 + xq satisfying the condition that q(x) ∼ Ai(x) as x → +∞ where Ai(x) is the Airy function. 3 The solution is unique and smooth, and is called the Hastings-McLeod solution [15,14]. The Tracy-Widom distributions are defined as [27,28] where The distribution functions in Definition 1.3 and 1.4 are more involved. Nevertheless they are expressible only in terms of q(x) above and two other functions a(x; w) and b(x; w). Let a(x; w) and b(x; w) be the solution to the initial value problem of the system of first order linear differential equations, There is a unique solution which is smooth in (x, w) ∈ R × C. The above differential equations are the first part of the Lax pair for the Painlevé II equation in the theory of integrable systems. Then (see (3.22) of [6]) F st (x; w + , w − ) = F 2 (x) a(x; w + )a(x; w − ) − a(x; w + )a(x; w − ) − b(x; w + )b(x; w − ) 2(w + + w − ) p(x) where p(x) := x ∞ q(y) 2 dy = q(x) 4 + xq(x) 2 − (q ′ (x)) 2 .