Moments Match between the KPZ Equation and the Airy Point Process

The results of Amir-Corwin-Quastel, Calabrese-Le Doussal-Rosso, Dotsenko, and Sasamoto-Spohn imply that the one-point distribution of the solution of the KPZ equation with the narrow wedge initial condition coincides with that for a multiplicative statistics of the Airy determinantal random point process. Taking Taylor coefficients of the two sides yields moment identities. We provide a simple direct proof of those via a combinatorial match of their multivariate integral representations.


Introduction
Since Tracy and Widom's discovery of the ASEP solvability eight years ago [18,19,20], the relationship between the "determinantal" and "non-determinantal" solvable models in the (1 + 1)dimensional KPZ (Kardar-Parisi-Zhang) universality class has largely remained a mystery. One step towards solving this mystery is the celebrated result of Amir-Corwin-Quastel [1], Calabrese-Le Doussal-Rosso [8], Dotsenko [10], and Sasamoto-Spohn [17], that provides an explicit expression for the distribution (or its Laplace transform) of one-point value of the solution of the KPZ equation with the so-called narrow wedge initial condition. It can be re-interpreted as saying that this Laplace transform coincides with the average of a multiplicative statistics of the Airy determinantal random point process. Although this restatement seems to be known to experts, we couldn't find it in this form in the literature, so we give an exact formulation as Theorem 2.1 below. Such a result is very useful as it immediately implies that this solution of the KPZ equation asymptotically at large times has the GUE Tracy-Widom distribution, which is a display of the KPZ universality, cf. Corwin's survey [9].
Finding other facts of similar nature has been a challenge so far. Imamura and Sasamoto [12] proved a similar statement for the O'Connell-Yor semi-discrete Brownian directed polymer. Unfortunately, the associated determinantal point process was not governed by a positive measure. Still, taking the edge limit of this process, they were able to recover Theorem 2.1. Another representation of the Laplace transform of the O'Connell-Yor partition function as the average of a multiplicative functional over a signed determinantal point process can be found in [14].
Very recently, one of the authors found in [4] an identity that relates a single point height distribution of the (higher spin inhomogeneous) stochastic six vertex model in a quadrant on one side, and multiplicative statistics of the Macdonald measures on the other. The ASEP limit of this identity was worked out in [7]. Taking the KPZ limit of both leads to Theorem 2.1 again.
The goal of this note is to look at Theorem 2.1 from the point of view of moments, rather than the corresponding distributions. One can study both the KPZ equation and the Airy point process via their exponential moments. Those are computationally tractable but they are of limited mathematical use because the corresponding moment problems are indeterminate. Still, on the KPZ side physicists were able to consistently use the moments to access the distributions via the (non-rigorous) replica trick, see [10] and [8] for early examples.
Our Theorem 2.2 proves the moments identity that corresponds to Theorem 2.1. The argument is a combinatorial match between known multivariate integral representations of the moments on both sides. Interestingly, these integral representations were known long before [1,8,10,17], but their similarity had not been exploited. We are hoping that the moments point of view will be beneficial for finding other similar correspondence.
We did attempt to extend the moments correspondence to a two-point identity, as integral representations on both side are again known. Unfortunately, we have not been successful in that so far.

The one-point equality
Let a 1 ≥ a 2 ≥ a 3 ≥ · · · be points of the Airy point process 1 at β = 2 (see, e.g., [2,11]) which is a determinantal point process on R with correlation kernel Here Ai(x) is the Airy function. From the opposite direction, let Z(T, X) denote the solution of the stochastic heat equation (see, e.g., [9,16]) whereẆ is the space-time white noise. H := − log(Z) is the Hopf-Cole solution of the Kardar-Parisi-Zhang stochastic partial differential equation with the narrow-edge initial data.
On the other hand, we could not find the following statement in the existing literature.
Our proof of Theorem 2.2 is based on direct comparison of contour integral formulas: for the right-hand side of (2) such formula is known as a solution for the attractive delta Bose gas equation, cf. the discussion in [6, Section 6.2], while for the left-hand side it can be computed through the Laplace transform of the correlation kernel K Airy . Remark 2.3. Expanding formally the result of Theorem 2.1 into power series in u and evaluating the coefficients, one gets the result of Theorem 2.2 and vice versa. However, these theorems are not equivalent: The u power series expansion of the left-hand side of (2.1) fails to converge for any u = 0. Below we provide two different proofs for Theorems 2.1 and 2.2, respectively.
The following corollary is present in [1,8,10,17], but it seems reasonable for us to give a proof using Theorem 2.1 above only, without appealing to the explicit evaluation of either side.
Equating (4) to (6) we are done. where On the other hand, the left-hand side of (1) is a multiplicative function of a determinantal point process and, therefore, also admits a Fredholm determinant formula, see, e.g., [3, equation (2.4)]: One immediately sees that upon the change of variables r i = −y i and identification T 2 = C 3 , the formulas (7) and (8) where the real numbers a 1 , . . . , a k satisfy a 1 a 2 · · · a k . It is convenient for us to modify the contours of integration in (9) to the imaginary axis iR. One collects certain residues in such a deformation, and the final result is read from [5, equation (13)] to be where λ = (λ 1 ≥ λ 2 ≥ . . . ) is a partition of k and (λ) is the number of non-zero parts λ j . Let us now produce a similar expression for the left-hand side of (2). Define the Laplace transform of the correlation functions of the Airy point process through The definition of the Airy point process implies that for a partition λ = (λ 1 ≥ λ 2 · · · ≥ λ ) = 1 m 1 2 m 2 · · · one has E m λ (exp(Ca 1 ), exp(Ca 2 ), . . . ) = 1 m 1 !m 2 ! · · · R(Cλ 1 , . . . , Cλ ), where m λ (y 1 , y 2 , . . . ) is the monomial symmetric function in variables y 1 , y 2 , . . . , as in [13,Chapter I]. Expanding h k into linear combination of m λ 's, we can then write where the summation goes over all partitions of k. Comparing (11) with (10), we see that it remains to identify the contour integrals over imaginary axis in (10) with R Cλ 1 , . . . , Cλ (λ) . The rest is based on the following identity that can be found in [15,Lemma 2.6]: Its immediate corollary is (we use an agreement z n+1 = z 1 and s n+1 = s 1 here, and also assume c 1 , . . . , c n > 0) Using the Gaussian integrals in variables z 1 , . . . , z n , the last formula is converted into Since i(z i − z i+1 ) has zero real part, we can integrate over s i in (12), arriving at the formula: We can now write the formula for R(c 1 , . . . , c n ) (we subdivide a permutation into cycles, use (13) and then combine back): .
Remark 3.1. We can use the Cauchy determinant formula with a i = −iz i + c i /2, b i = iz i + c i /2 to simplify the last determinant.
We now take a partition λ k with (λ) = n, set c i = Cλ i and make a change of variables iz j = Cw j + Cλ j 2 − C 2 to get (note that we deformed the contours to the imaginary axis; we do not pick up any residues in such a deformation) R(Cλ 1 , . . . , Cλ n ) = exp C 3 . (14) It remains to simplify the exponents: Combining (11) with (14), (15) and identifying C 3 = T 2 . we arrive at (10) multiplied by exp(−kT /24).