A System of Differential Equations for the Airy Process

A PDE for the multi-time joint probability of the Airy process

This paper gives a PDE for multi-time joint probability of the Airy process, which generalizes Adler and van Moerbeke's result on the 2-time case. As an intermediate step, the PDE for the multi-time joint probability of the Dyson Brownian motion is also given.

The Riemann-Hilbert approach to the generating function of the higher order Airy point processes

We prove a Tracy-Widom type formula for the generating function of occupancy numbers on several disjoint intervals of the higher order Airy point processes. The formula is related to a new vector-valued Painlevé II hierarchy we define, together with its Lax pair.

Gap probabilities in the bulk of the Airy process

We consider the probability that no points lie on $g$ large intervals in the bulk of the Airy point process. We make a conjecture for all the terms in the asymptotics up to and including the oscillations of order $1$, and we prove this conjecture for $g=1$.

The half-space Airy stat process

We study the multipoint distribution of stationary half-space last passage percolation with exponentially weighted times. We derive both finite-size and asymptotic results for this distribution. In the latter case we observe a new one-parameter process we call half-space Airy stat. It is a one-parameter generalization of the Airy stat process of Baik-Ferrari-Péché, which is recovered far away from the diagonal. All these results extend the one-point results previously proven by the authors.

The Airy$_2$ process and the 3D Ising model

The Ferrari-Spohn diffusion process arises as limit process for the 2D Ising model as well as random walks with area penalty. Motivated by the 3D Ising model, we consider $M$ such diffusions conditioned not to intersect. We show that the top process converges to the Airy$_2$ process as $M\to\infty$. We then explain the relation with the 3D Ising model and present some conjectures about it.

The gap probabilities of the tacnode, Pearcey and Airy point processes, their mutual relationship and evaluation

We express the gap probabilities of the tacnode process as the ratio of two Fredholm determinants; the denominator is the standard Tracy-Widom distribution, while the numerator is the Fredholm determinant of a very explicit kernel constructed with Airy functions and exponentials. The formula allows us to apply the theory of numerical evaluation of Fredholm determinants and thus produce numerical results for the gap probabilities. In particular we investigate numerically how, in different regimes, the Pearcey process degenerates to the Airy one, and the tacnode degenerates to the Pearcey and Ai…

Momenta spacing distributions in anharmonic oscillators and the higher order finite temperature Airy kernel

We rigorously compute the integrable system for the limiting $(N\rightarrow\infty)$ distribution function of the extreme momentum of $N$ noninteracting fermions when confined to an anharmonic trap $V(q)=q^{2n}$ for $n\in\mathbb{Z}_{\geq 1}$ at positive temperature. More precisely, the edge momentum statistics in the harmonic trap $n=1$ are known to obey the weak asymmetric KPZ crossover law which is realized via the finite temperature Airy kernel determinant or equivalently via a Painlevé-II integro-differential transcendent, cf. \cite{LW,ACQ}. For general $n\geq 2$, a novel higher order finit…

From the Pearcey to the Airy process

Putting dynamics into random matrix models leads to finitely many nonintersecting Brownian motions on the real line for the eigenvalues, as was discovered by Dyson. Applying scaling limits to the random matrix models, combined with Dyson's dynamics, then leads to interesting, infinite-dimensional diffusions for the eigenvalues. This paper studies the relationship between two of the models, namely the Airy and Pearcey processes and more precisely shows how to approximate the multi-time statistics for the Pearcey process by the one of the Airy process with the help of a PDE governing the gap pro…

Transition between Airy_1 and Airy_2 processes and TASEP fluctuations

We consider the totally asymmetric simple exclusion process, a model in the KPZ universality class. We focus on the fluctuations of particle positions starting with certain deterministic initial conditions. For large time t, one has regions with constant and linearly decreasing density. The fluctuations on these two regions are given by the Airy_1 and Airy_2 processes, whose one-point distributions are the GOE and GUE Tracy-Widom distributions of random matrix theory. In this paper we analyze the transition region between these two regimes and obtain the transition process. Its one-point distr…

Airy point process at the liquid-gas boundary

Domino tilings of the two-periodic Aztec diamond feature all of the three possible types of phases of random tiling models. These phases are determined by the decay of correlations between dominoes and are generally known as solid, liquid and gas. The liquid-solid boundary is easy to define microscopically and is known in many models to be described by the Airy process in the limit of a large random tiling. The liquid-gas boundary has no obvious microscopic description. Using the height function we define a random measure in the two-periodic Aztec diamond designed to detect the long range corr…

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.

The arctic circle boundary and the Airy process

We prove that the, appropriately rescaled, boundary of the north polar region in the Aztec diamond converges to the Airy process. The proof uses certain determinantal point processes given by the extended Krawtchouk kernel. We also prove a version of Propp's conjecture concerning the structure of the tiling at the center of the Aztec diamond.

Airy process at a thin rough region between frozen and smooth

We show there is a last path at the rough smooth boundary of the two-periodic Aztec diamond with parameter $a\in (0,1)$ that, suitably rescaled, converges to the Airy process, under the condition that $a$ tends to zero as the size of the Aztec diamond tends to infinity at a certain rate. This condition causes the rough region to have a thin, mesoscopic width. We also show that the dimers are described by a discrete Bessel kernel when the width is only of microscopic size.

Supremum of the Airy2 process minus a parabola on a half line

Let $\aip(t)$ be the Airy$_2$ process. We show that the random variable [\sup_{t\leqα}\{aip(t)-t^2}+\min{0,α}^2] has the same distribution as the one-point marginal of the Airy$_{2\to1}$ process at time $α$. These marginals form a family of distributions crossing over from the GUE Tracy-Widom distribution $F_{\rm GUE}(x)$ for the Gaussian Unitary Ensemble of random matrices, to a rescaled version of the GOE Tracy-Widom distribution $F_{\rm GOE}(4^{1/3}x)$ for the Gaussian Orthogonal Ensemble. Furthermore, we show that for every $α$ the distribution has the same right tail decay $e^{-(4/3)x^{3/…

PDEs for the joint distributions of the Dyson, Airy and Sine processes

In a celebrated paper, Dyson shows that the spectrum of an n\times n random Hermitian matrix, diffusing according to an Ornstein-Uhlenbeck process, evolves as n noncolliding Brownian motions held together by a drift term. The universal edge and bulk scalings for Hermitian random matrices, applied to the Dyson process, lead to the Airy and Sine processes. In particular, the Airy process is a continuous stationary process, describing the motion of the outermost particle of the Dyson Brownian motion, when the number of particles gets large, with space and time appropriately rescaled. In this pape…

Chernoff's distribution and differential equations of parabolic and Airy type

We give a direct derivation of the distribution of the maximum and the location of the maximum of one-sided and two-sided Brownian motion with a negative parabolic drift. The argument uses a relation between integrals of special functions, in particular involving integrals with respect to functions which can be called "incomplete Scorer functions". The relation is proved by showing that both integrals, as a function of two parameters, satisfy the same extended heat equation, and the maximum principle is used to show that these solution must therefore have the stated relation. Once this relatio…

On the exponent governing the correlation decay of the Airy$_1$ process

We study the decay of the covariance of the Airy$_1$ process, $\mathcal{A}_1$, a stationary stochastic process on $\mathbb{R}$ that arises as a universal scaling limit in the Kardar-Parisi-Zhang (KPZ) universality class. We show that the decay is super-exponential and determine the leading order term in the exponent by showing that $\textrm{Cov}(\mathcal{A}_1(0),\mathcal{A}_1(u))= e^{-(\frac{4}{3}+o(1))u^3}$ as $u\to\infty$. The proof employs a combination of probabilistic techniques and integrable probability estimates. The upper bound uses the connection of $\mathcal{A}_1$ to planar exponent…

Airy Point Process via Supersymmetric Lifts

We study the local asymptotics at the edge for particle systems arising from: (i) eigenvalues of sums of unitarily invariant random Hermitian matrices and (ii) signatures corresponding to decompositions of tensor products of representations of the unitary group. Our method treats these two models in parallel, and is based on new formulas for observables described in terms of a special family of lifts, which we call supersymmetric lifts, of Schur functions and multivariate Bessel functions. We obtain explicit expressions for a class of supersymmetric lifts inspired by determinantal formulas for…

Airy 点过程的大偏差界

Large deviation bounds for the Airy point process

本文首次建立 Airy 点过程的大偏差界，通过 GUE 近似、随机 Airy 算子估计和 beta 系综边缘刚性，统一改进了 KPZ 方程及半空间 KPZ 方程的下尾概率估计，并得到尖锐超指数衰减。

On Asymptotics for the Airy Process

The Airy process A(t), introduced by Prähofer and Spohn, is the limiting stationary process for a polynuclear growth model. Adler and van Moerbeke found a PDE in the variables s_1, s_2, and t for the probability that A(0)<s_1 and A(t)<s_2. Using this they were able, assuming the truth of a certain conjecture and appropriate uniformity, to obtain the first few terms of an asymptotic expansion for this probability as t->infinity, with fixed s_1 and s_2. We shall show that the expansion can be obtained by using the Fredholm determinant representation for the probability. The main ingredients are …

高斯β-ensemble特征多项式的强逼近：边缘区域与随机Airy函数

Strong approximation of Gaussian $β$-ensemble characteristic polynomials: the edge regime and the stochastic Airy function

该研究通过转移矩阵递推分析高斯β-ensemble特征多项式，证明其重标后在谱边缘收敛到随机Airy函数，建立耦合关系并得到均匀逼近速率，成果基于前期双曲部分分析。

Rigidity of Determinantal Point Processes with the Airy, the Bessel and the Gamma Kernel

A point process is said to be rigid if for any bounded domain in the phase space, the number of particles in the domain is almost surely determined by the restriction of the configuration to the complement of our bounded domain. The main result of this paper is that determinantal point processes with the Airy, the Bessel and the Gamma kernels are rigid. The proof follows the scheme of Ghosh [6], Ghosh and Peres [7]: the main step is the construction of a sequence of additive statistics with variance going to zero.

Airy$_1$过程中持续概率的精确衰减

Exact decay of the persistence probability in the Airy$_1$ process

本研究针对Airy$_1$过程的持续概率衰减问题，推导了阈值$c\neq 3/2$时衰减速率$κ(c)$的解析表达式，完成了解析延拓并数值验证了$c<0$时的有效性。

Boundary Asymptotics of Non-Intersecting Brownian Motions: Pearcey, Airy and a Transition

We study $n$ non-intersecting Brownian motions, corresponding to the eigenvalues of an $n\times n$ Hermitian Brownian motion. At the boundary of their limit shape we find that only three universal processes can arise: the Pearcey process close to merging points, the Airy line ensemble at edges and a novel determinantal process describing the transition from the Pearcey process to the Airy line ensemble. The three cases are distinguished by a remarkably simple integral condition. Our results hold under very mild assumptions, in particular we do not require any kind of convergence of the initial…

Airy processes with wanderers and new universality classes

Consider $n+m$ nonintersecting Brownian bridges, with $n$ of them leaving from 0 at time $t=-1$ and returning to 0 at time $t=1$, while the $m$ remaining ones (wanderers) go from $m$ points $a_i$ to $m$ points $b_i$. First, we keep $m$ fixed and we scale $a_i,b_i$ appropriately with $n$. In the large-$n$ limit, we obtain a new Airy process with wanderers, in the neighborhood of $\sqrt{2n}$, the approximate location of the rightmost particle in the absence of wanderers. This new process is governed by an Airy-type kernel, with a rational perturbation. Letting the number $m$ of wanderers tend to…

On the joint distribution of the maximum and its position of the Airy2 process minus a parabola

The maximal point of the Airy2 process minus a parabola is believed to describe the scaling limit of the end-point of the directed polymer in a random medium, which was proved to be true for a few specific cases. Recently two different formulas for the joint distribution of the location and the height of this maximal point were obtained, one by Moreno Flores, Quastel and Remenik, and the other by Schehr. The first formula is given in terms of the Airy function and an associated operator, and the second formula is expressed in terms of the Lax pair equations of the Painleve II equation. We give…

Brownian Gibbs property for Airy line ensembles

Consider N Brownian bridges B_i:[-N,N] -> R, B_i(-N) = B_i(N) = 0, 1 <= i <= N, conditioned not to intersect. The edge-scaling limit of this system is obtained by taking a limit as N -> infinity of these curves scaled around (0,2^{1/2} N) horizontally by a factor of N^{2/3} and vertically by N^{1/3}. If a parabola is added to each limit curve, an x-translation invariant process sometimes called the multi-line Airy process is obtained. We prove the existence of a version of this process (which we call the Airy line ensemble) in which curves are a.s. everywhere continuous and non-intersecting. T…

Two-point convergence of the stochastic six-vertex model to the Airy process

In this paper we consider the stochastic six-vertex model in the quadrant started with step initial data. After a long time $T$, it is known that the one-point height function fluctuations are of order $T^{1/3}$ and governed by the Tracy-Widom distribution. We prove that the two-point distribution of the height function, rescaled horizontally by $T^{2/3}$ and vertically by $T^{1/3}$, converges to the two-point distribution of the Airy process. The starting point of this result is a recent connection discovered by Borodin-Bufetov-Wheeler between the stochastic six-vertex model and the ascending…

Airy$_1$过程的遍历性、中心极限定理与渐近最大值

Ergodicity, CLT and asymptotic maximum of the Airy$_1$ process

该研究利用随机热方程解的关联性质及KPZ方程收敛性证明Airy$_1$过程的关联性，通过Newman不等式建立其遍历性和中心极限定理，推导泊松极限定理并精确估计区间最大值的渐近行为，同时给出Airy$_2$过程的类似结果。

对Ferrari与Liu论文《Airy$_1$过程中持续概率的精确衰减》的评论

Comment on the paper "Exact decay of the persistence probability in the Airy$_1$ process" by Ferrari and Liu

本评论指出Ferrari和Liu得到的Airy$_1$过程持续概率非平凡函数，与全不对称排斥过程流涨落大偏差函数形式相似，并解释二者均对应系统维持高于典型值流的同类稀有事件条件约束。

The generating function for the Airy point process and a system of coupled Painlevé II equations

For a wide class of Hermitian random matrices, the limit distribution of the eigenvalues close to the largest one is governed by the Airy point process. In such ensembles, the limit distribution of the k-th largest eigenvalue is given in terms of the Airy kernel Fredholm determinant or in terms of Tracy-Widom formulas involving solutions of the Painlevé II equation. Limit distributions for quantities involving two or more near-extreme eigenvalues, such as the gap between the k-th and the \ell-th largest eigenvalue or the sum of the k largest eigenvalues, can be expressed in terms of Fredholm d…

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.

Asymptotics for the Covariance of the Airy_2 process

In this paper we compute some of the higher order terms in the large-t asymptotic expansion of the Airy process two-point function, extending the previous work of Adler and van Moerbeke and Widom. We prove that it is possible to represent any order asymptotic approximation as a polynomial and integrals of the Hastings-McLeod Painlevé II function and its first derivative. Further, for up to tenth order we give this asymptotic approximation as a linear combination of the Tracy-Widom GUE density function f_2 and its derivatives. As a corollary to this, the asymptotic covariance is expressed up to…

广义Airy型方程、热型方程与伪过程

General Airy-type equations, heat-type equations and pseudo-processes

该研究系统分析高阶Airy型方程的解、展开式与概率解释，推导相关微分方程，将高阶热型分数阶Cauchy问题基本解用Airy型函数表示，并借助伪过程理论推广结果、研究解的矩。

The Cusp-Airy Process

At a typical cusp point of the disordered region in a random tiling model we expect to see a determinantal process called the Pearcey process in the appropriate scaling limit. However, in certain situations another limiting point process appears that we call the Cusp-Airy process, which is a kind of two sided extension of the Airy kernel point process. We will study this problem in a class of random lozenge tiling models coming from interlacing particle systems. The situation was briefly studied previously by Okounkov and Reshetikhin under the name cuspidal turning point.

Airy过程水平集的宏观Hausdorff维数

Macroscopic Hausdorff dimension of the level sets of the Airy processes

本文研究Airy$_1$和Airy$_2$过程水平集的宏观Hausdorff维数，分别借助联合概率不等式与极值尾部概率的定量估计，扩展了Khoshnevisan等人的一般方法。

Jack过程与拓扑展开的艾里极限

Airy limit for the Jack process and topological expansion

该研究建立Jack-Plancherel过程的多时间软边矩收敛性，推导其极限矩的布朗桥展开，确定等时极限与Airy_β过程的关联，得到边际b猜想的渐近β拓扑展开及β=2时Witten-Kontsevich相交数的显式非负公式。

关于抛物 Airy 过程的多点分布公式

On the multipoint distribution formulas of the parabolic Airy process

本文直接证明了 arXiv:1907.09876 给出的抛物 Airy 过程等时间多点分布公式与扩展 Airy 核 Fredholm 行列式公式等价，并得到新分布公式及 Andreief 恒等式推广。

GOE and ${\rm Airy}_{2\to 1}$ marginal distribution via symplectic Schur functions

We derive Sasamoto's Fredholm determinant formula for the Tracy-Widom GOE distribution, as well as the one-point marginal distribution of the ${\rm Airy}_{2\to1}$ process, originally derived by Borodin-Ferrari-Sasamoto, as scaling limits of point-to-line and point-to-half-line last passage percolation with exponentially distributed waiting times. The asymptotic analysis goes through new expressions for the last passage times in terms of integrals of (the continuous analog of) symplectic and classical Schur functions, obtained recently in [BZ19a].

Periodic Airy process and equilibrium dynamics of edge fermions in a trap

We establish an exact mapping between (i) the equilibrium (imaginary time) dynamics of non-interacting fermions trapped in a harmonic potential at temperature $T=1/β$ and (ii) non-intersecting Ornstein-Uhlenbeck (OU) particles constrained to return to their initial positions after time $β$. Exploiting the determinantal structure of the process we compute the universal correlation functions both in the bulk and at the edge of the trapped Fermi gas. The latter corresponds to the top path of the non-intersecting OU particles, and leads us to introduce and study the time-periodic Airy$_2$ process,…

Scale Invariance of the PNG Droplet and the Airy Process

We establish that the static height fluctuations of a particular growth model, the PNG droplet, converges upon proper rescaling to a limit process, which we call the Airy process A(y). The Airy process is stationary, it has continuous sample paths, its single "time" (fixed y) distribution is the Tracy-Widom distribution of the largest eigenvalue of a GUE random matrix, and the Airy process has a slow decay of correlations as y^(-2). Roughly the Airy process describes the last line of Dyson's Brownian motion model for random matrices. Our construction uses a multi-layer version of the PNG model…

Evolution of the Stochastic Airy eigenvalues under a changing boundary

The Airy$_β$ point process, originally introduced by Ramírez, Rider, and Virág, is defined as the spectrum of the stochastic Airy operator $\mathcal{H}_β$ acting on a subspace of $L^2[0,\infty)$ with Dirichlet boundary condition. In this paper we study the coupled family of point processes defined as the eigenvalues of $\mathcal{H}_β$ acting on a subspace of $L^2[t,\infty)$. These point processes are coupled through the Brownian term of $\mathcal{H}_β$. We show that these point processes as a function of $t$ are differentiable with explicitly computable derivative. Moreover when recentered by …

The stochastic Airy operator at large temperature

It was shown in [J. A. Ramírez, B. Rider and B. Virág. J. Amer. Math. Soc. 24, 919-944 (2011)] that the edge of the spectrum of $β$ ensembles converges in the large $N$ limit to the bottom of the spectrum of the stochastic Airy operator. In the present paper, we obtain a complete description of the bottom of this spectrum when the temperature $1/β$ goes to $\infty$: we show that the point process of appropriately rescaled eigenvalues converges to a Poisson point process on $\mathbb{R}$ of intensity $e^x dx$ and that the eigenfunctions converge to Dirac masses centered at IID points with expone…

Discrete polynuclear growth and determinantal processes

We consider a discrete polynuclear growth (PNG) process and prove a functioal limit theorem for its convergrence to the Airy process. This generalizes previous results by Pr"ahofer and Spohn. The result enables us to express the GOE largest eigenvalue (Tracy-Widom) distribution in terms of the Airy process. We also show some results and give a conjecture about the transversal fluctuations in a point to line last passage percolation problem.

Pfaffian点过程到Airy线集的均匀收敛

Uniform convergence of Pfaffian point process to the Airy line ensemble

本文研究了Pfaffian Schur过程到Airy线集的均匀收敛性，并通过分析核的渐进行为验证了收敛条件。

Investigation of Airy equations with random initial conditions

The paper investigates properties of mean-square solutions to the Airy equation with random initial data given by stationary processes. The result on the modulus of continiuty of the solution is stated and properties of the covariance function are described. Bounds for the distributions of the suprema of solutions under $φ$-sub-Gaussian initial conditions are presented. Several examples are provided to illustrate the results. Extension of the results to the case of fractional Airy equation is given.

Stable distributions and pseudo-processes related to fractional Airy functions

In this paper we study pseudo-processes related to odd-order heat-type equations composed with Lévy stable subordinators. The aim of the article is twofold. We first show that the pseudo-density of the subordinated pseudo-process can be represented as an expectation of damped oscillations with generalized gamma distributed parameters. This stochastic representation also arises as the solution to a fractional diffusion equation, involving a higher-order Riesz-Feller operator, which generalizes the odd-order heat-type equation. We then prove that, if the stable subordinator has a suitable expone…

带漫游者的 Airy 过程、KPZ 涨落以及 Tracy–Widom GOE 分布的一种形变

Airy process with wanderers, KPZ fluctuations, and a deformation of the Tracy--Widom GOE distribution

本文研究带 m 个漫游者的 Airy 过程上确界分布，证明其为 Tracy–Widom GOE 分布的 2m 参数形变，并给出 Fredholm 行列式、Painlevé II、KPZ 不动点、PDE 与 KdV 等多种刻画。

Dyson's non-intersecting Brownian motions with a few outliers

Consider n non-intersecting particles on the real line (Dyson Brownian motions), all starting from the origin at time=0, and forced to return to x=0 at time=1. For large n, the average mean density of particles has its support, for each 0<t<1, within the interior of an ellipse. The Airy process is defined as the motion of these non-intersecting Brownian motions for large n, but viewed from an arbitrary point on the ellipse with an appropriate space-time rescaling. Assume now a finite number r of these particles are forced to a different target point. Does it affect the Brownian fluctuations al…

Markov property of determinantal processes with extended sine, Airy, and Bessel kernels

When the number of particles is finite, the noncolliding Brownian motion (the Dyson model) and the noncolliding squared Bessel process are determinantal diffusion processes for any deterministic initial configuration $ξ=\sum_{j \in Λ} δ_{x_j}$, in the sense that any multitime correlation function is given by a determinant associated with the correlation kernel, which is specified by an entire function $Φ$ having zeros in $\supp ξ$. Using such entire functions $Φ$, we define new topologies called the $Φ$-moderate topologies. Then we construct three infinite-dimensional determinantal processes, …

Differential Equations for Dyson Processes

We call "Dyson process" any process on ensembles of matrices in which the entries undergo diffusion. We are interested in the distribution of the eigenvalues (or singular values) of such matrices. In the original Dyson process it was the ensemble of n by n Hermitian matrices, and the eigenvalues describe n curves. Given sets X_1,...,X_m the probability that for each k no curve passes through X_k at time τ_k is given by the Fredholm determinant of a certain matrix kernel, the extended Hermite kernel. For this reason we call this Dyson process the Hermite process. Similarly, when the entries of …

Multiplicative functionals on ensembles of non-intersecting paths

The purpose of this article is to develop a theory behind the occurrence of "path-integral" kernels in the study of extended determinantal point processes and non-intersecting line ensembles. Our first result shows how determinants involving such kernels arise naturally in studying ratios of partition functions and expectations of multiplicative functionals for ensembles of non-intersecting paths on weighted graphs. Our second result shows how Fredholm determinants with extended kernels (as arise in the study of extended determinantal point processes such as the Airy_2 process) are equal to Fr…

Characterization of Brownian Gibbsian line ensembles

In this paper we show that a Brownian Gibbsian line ensemble is completely characterized by the finite-dimensional marginals of its top curve, i.e. the finite-dimensional sets of the its top curve form a separating class. A particular consequence of our result is that the Airy line ensemble is the unique Brownian Gibbsian line ensemble, whose top curve is the Airy$_2$ process.

Exceptional times when the KPZ fixed point violates Johansson's conjecture on maximizer uniqueness

In 2002, Johansson conjectured that the maximum of the Airy$_2$ process minus the parabola $x^2$ is almost surely achieved at a unique location. This result was proved a decade later by Corwin and Hammond; Moreno Flores, Quastel and Remenik; and Pimentel. Up to scaling, the Airy$_2$ process minus the parabola $x^2$ arises as the fixed time spatial marginal of the KPZ fixed point when started from narrow wedge initial data. We extend this maximizer uniqueness result to the fixed time spatial marginal of the KPZ fixed point when begun from any element of a very broad class of initial data. None …