Elliot Blackstone, Christophe Charlier, Jonatan Lenells
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 $g=1$.
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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…
We obtain a formula for the $n$-dimensional distributions of the Airy$_1$ process in terms of a Fredholm determinant on $L^2(\rr)$, as opposed to the standard formula which involves extended kernels, on $L^2(\{1,...,n\}\times\rr)$. The formula is analogous to an earlier formula of [PS02] for the Airy$_2$ process. Using this formula we are able to prove that the Airy$_1$ process is Hölder continuous with exponent $\frac12-$ and that it fluctuates locally like a Brownian motion. We also explain how the same methods can be used to obtain the analogous results for the Airy$_2$ process. As a conseq…
The Airy line ensemble is a central object in random matrix theory and last passage percolation defined by a determinantal formula. The goal of this paper is to provide a set of tools which allow for precise probabilistic analysis of the Airy line ensemble. The two main theorems are a representation in terms of independent Brownian bridges connecting a fine grid of points, and a modulus of continuity result for all lines. Along the way, we give tail bounds and moduli of continuity for nonintersecting Brownian ensembles, and a quick proof of tightness for Dyson's Brownian motion converging to t…
We consider several aspects of the scaling limit of percolation on random planar triangulations, both finite and infinite. The equivalents for random maps of Cardy's formula for the limit under scaling of various crossing probabilities are given. The limit probabilities are expressed in terms of simple events regarding Airy-Levy processes. Some explicit formulas for limit probabilities follow from this relation by applying known results on stable processes. Conversely, natural symmetries of the random maps imply identities concerning the Airy-Levy processes.
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…
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。
Airy and Pearcey-like kernels and generalizations arising in random matrix theory are expressed as double integrals of ratios of exponentials, possibly multiplied with a rational function. In this work it is shown that such kernels are intimately related to wave functions for polynomial (Gel'fand-Dickey reductions) or rational reductions of the KP-hierarchy; their Fredholm determinant also satisfies linear PDEs (Virasoro constraints), yielding, in a systematic way, non-linear PDEs for the Fredholm determinant of such kernels. Examples include Fredholm determinants giving the gap probability of…
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…
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…
The Airy line ensemble at the edge of uniform alternating sign matrices
本文证明均匀交错符号矩阵的高度路径系在北极弧内点收敛于Airy线系，利用Pfaffian公式和重采样定律，结合Brownian Gibbs性质与强刻画定理。
We consider the system of one-sided reflected Brownian motions which is in variational duality with Brownian last passage percolation. We show that it has integrable transition probabilities, expressed in terms of Hermite polynomials and hitting times of exponential random walks, and that it converges in the 1:2:3 scaling limit to the KPZ fixed point, the scaling invariant Markov process defined in [MQR17] and believed to govern the long time large scale fluctuations for all models in the KPZ universality class. Brownian last passage percolation was shown recently in [DOV18] to converge to the…
An explicit Fredholm determinant formula is derived for the multipoint distribution of the height function of the totally asymmetric simple exclusion process (TASEP) with arbitrary right-finite initial condition. The method is by solving the biorthogonal ensemble/non-intersecting path representation found by [Sas05; BFPS07]. The resulting kernel involves transition probabilities of a random walk forced to hit a curve defined by the initial data. In the KPZ 1:2:3 scaling limit the formula leads in a transparent way to a Fredholm determinant formula, in terms of analogous kernels based on Browni…
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,…
Operator level soft edge to bulk transition in $β$-ensembles via canonical systems
本文研究了β-集合中软边缘到体极限的运算符层面转换，通过canonical systems统一框架证明Airy算子在高能尺度下收敛到sine算子，并推导了相关Weyl-Titchmarsh函数和谱测度的收敛。
We present a method to derive asymptotics of eigenvalues for trace-class integral operators $K:L^2(J;dλ)\circlearrowleft$, acting on a single interval $J\subset\mathbb{R}$, which belong to the ring of integrable operators \cite{IIKS}. Our emphasis lies on the behavior of the spectrum $\{λ_i(J)\}_{i=0}^{\infty}$ of $K$ as $|J|\rightarrow\infty$ and $i$ is fixed. We show that this behavior is intimately linked to the analysis of the Fredholm determinant $\det(I-γK)|_{L^2(J)}$ as $|J|\rightarrow\infty$ and $γ\uparrow 1$ in a Stokes type scaling regime. Concrete asymptotic formulæ\, are obtained f…
Loop Equations Characterize Random Matrix Statistics
研究证明随机矩阵理论的通用局部点过程由循环方程层次结构刻画，如正弦β点过程是体循环方程唯一解，艾里β点过程是边缘循环方程唯一解，此为通用性提供直接途径，许多模型中方程可由局部定律和分部积分得出。
We prove a local central limit theorem (LCLT) for the number of points $N(J)$ in a region $J$ in $\mathbb R^d$ specified by a determinantal point process with an Hermitian kernel. The only assumption is that the variance of $N(J)$ tends to infinity as $|J| \to \infty$. This extends a previous result giving a weaker central limit theorem (CLT) for these systems. Our result relies on the fact that the Lee-Yang zeros of the generating function for $\{E(k;J)\}$ --- the probabilities of there being exactly $k$ points in $J$ --- all lie on the negative real $z$-axis. In particular, the result applie…
One-arm probabilities for the two-dimensional metric-graph and discrete Gaussian free field
研究离散和度量图高斯自由场在盒状区域狄利克雷边界条件下的单臂概率，推导度量图的渐进行为并比较离散与度量图的点到体概率差异。
We study probabilities of various rare events for the limiting point process that appears at the random matrix hard edge. We also show a transition from hard edge to bulk behavior. Asymptotic events studied include a central limit theorem and large deviation result for the number of points in a growing interval $[0,λ]$ as $λ\to \infty$. We study these results for the square root of the hard edge process. In this setting many of these behaviors mimic those of the Sine$_β$ process.
This paper presents a novel method for computing certain particle locations in the multi-species $q$-TAZRP (totally asymmetric zero range process). The method is based on a decomposition of the process into its discrete-time embedded Markov chain, which is described more generally as a monotone process on a graded partially ordered set; and an independent family of exponential random variables. A further ingredient is explicit contour integral formulas for the transition probabilities of the $q$-TAZRP. The main result of this method is a shift invariance for the multi-species $q$-TAZRP on the …
We consider n one-dimensional Brownian motions, such that n/2 Brownian motions start at time t=0 in the starting point a and end at time t=1 in the endpoint b and the other n/2 Brownian motions start at time t=0 at the point -a and end at time t=1 in the point -b, conditioned that the n Brownian paths do not intersect in the whole time interval (0,1). The correlation functions of the positions of the non-intersecting Brownian motions have a determinantal form with a kernel that is expressed in terms of multiple Hermite polynomials of mixed type. We analyze this kernel in the large n limit for …
We study two one-parameter families of point processes connected to random matrices: the Sine_beta and Sch_tau processes. The first one is the bulk point process limit for the Gaussian beta-ensemble. For beta=1, 2 and 4 it gives the limit of the GOE, GUE and GSE models of random matrix theory. In particular, for beta=2 it is a determinantal point process conjectured to have similar behavior to the critical zeros of the Riemann zeta-function. The second process can be obtained as the bulk scaling limit of the spectrum of certain discrete one-dimensional random Schrödinger operators. Both proces…
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