Airy Point Process Research Articles

Wiener densities for the Airy line ensemble

AbstractThe parabolic Airy line ensemble is a central limit object in the KPZ (Kardar–Parisi–Zhang) universality class and related areas. On any compact set , the law of the recentered ensemble has a density with respect to the law of independent Brownian motions. We show where is an explicit, tractable, non‐negative function of . We use this formula to show that is bounded above by a ‐dependent constant, give a sharp estimate on the size of the set where as , and prove a large deviation principle for . We also give density estimates that take into account the relative positions of the Airy lines, and prove sharp two‐point tail bounds that are stronger than those for Brownian motion. These estimates are a key input in the classification of geodesic networks in the directed landscape. The paper is essentially self‐contained, requiring only tail bounds on the Airy point process and the Brownian Gibbs property as inputs.

The complex elliptic Ginibre ensemble at weak non-Hermiticity: Edge spacing distributions

The focus of this paper is on the distribution function of the rightmost eigenvalue for the complex elliptic Ginibre ensemble in the limit of weak non-Hermiticity. We show how the limiting distribution function can be expressed in terms of an integro-differential Painlevé-II function and how the same captures the nontrivial transition between Poisson and Airy point process extreme value statistics as the degree of non-Hermiticity decreases. Our most explicit new results concern the tail asymptotics of the limiting distribution function. For the right tail we compute the leading order asymptotics uniformly in the degree of non-Hermiticity, for the left tail we compute it close to Hermiticity.

Integrable equations associated with the finite‐temperature deformation of the discrete Bessel point process

AbstractWe study the finite‐temperature deformation of the discrete Bessel point process. We show that its largest particle distribution satisfies a reduction of the 2D Toda equation, as well as a discrete version of the integro‐differential Painlevé II equation of Amir–Corwin–Quastel, and we compute initial conditions for the Poissonization parameter equal to 0. As proved by Betea and Bouttier, in a suitable continuum limit the last particle distribution converges to that of the finite‐temperature Airy point process. We show that the reduction of the 2D Toda equation reduces to the Korteweg–de Vries equation, as well as the discrete integro‐differential Painlevé II equation reduces to its continuous version. Our approach is based on the discrete analogue of Its–Izergin–Korepin–Slavnov theory of integrable operators developed by Borodin and Deift.

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 supersymmetric Schur functions due to \cite{MJ03}. Asymptotic analysis of these lifts enable us to probe the edge. We focus on several settings where the Airy point process arises.

Exact lower-tail large deviations of the KPZ equation

Consider the Hopf–Cole solution h(t,x) of the Kandar–Parisi–Zhang (KPZ) equation with the narrow wedge initial condition. Regarding t→∞ as a scaling parameter, we provide the first rigorous proof of the large deviation principle (LDP) for the lower tail of h(2t,0)+t 12, with speed t2 and an explicit rate function Φ −(z). This result confirms existing physics predictions made by Corwin (2011); Sasorov, Meerson, and Prolhac (2017); and Krajenbrink, Le Doussal, and Prolhac (2018). Our analysis utilizes a formula from Borodin and Gorin (2016) to convert the LDP for the KPZ equation to calculating an exponential moment of the Airy point process (PP). To estimate this exponential moment, we invoke the stochastic Airy operator (SAO) and use the Riccati transform, comparison techniques, and certain variational characterizations of the relevant functional.

Fluctuations of $$\beta $$-Jacobi product processes

We study Markov chains formed by squared singular values of products of truncated orthogonal, unitary, symplectic matrices (corresponding to the Dyson index $$\beta = 1,2,4$$ respectively) where time corresponds to the number of terms in the product. More generally, we consider the $$\beta $$ -Jacobi product process obtained by extrapolating to arbitrary $$\beta > 0$$ . For fixed time (i.e. number of factors is constant), we show that the global fluctuations are jointly Gaussian with explicit covariances. For time growing linearly with matrix size, we show convergence of moments after suitable rescaling. When $$\beta = 2$$ , our results imply that the right edge converges to a process which interpolates between the Airy point process and a deterministic configuration. This process connects a time-parametrized family of point processes appearing in the works of Akemann–Burda–Kieburg and Liu–Wang–Wang across time. In the arbitrary $$\beta > 0$$ case, our results show tightness of the particles near the right edge. The limiting moment formulas correspond to expressions for the Laplace transform of a conjectural $$\beta $$ -generalization of the interpolating process.

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.

Global rigidity and exponential moments for soft and hard edge point processes

We establish sharp global rigidity upper bounds for universal determinantal point processes describing edge eigenvalues of random matrices. For this, we first obtain a general result which can be applied to general (not necessarily determinantal) point processes which have a smallest (or largest) point: it allows to deduce global rigidity upper bounds from the exponential moments of the counting function of the process. By combining this with known exponential moment asymptotics for the Airy and Bessel point processes, we improve on the best known upper bounds for the global rigidity of the Airy point process, and we obtain new global rigidity results for the Bessel point process. Secondly, we obtain exponential moment asymptotics for the Wright's generalized Bessel process and the Meijer-$\mathrm{G}$ process, up to and including the constant term. As a direct consequence, we obtain new results for the expectation and variance of the associated counting functions. Furthermore, by combining these asymptotics with our general rigidity theorem, we obtain new global rigidity upper bounds for these point processes.

Large deviations for the Kardar– Parisi–Zhang equation from the Kadomtsev–Petviashvili equation

Recently, Quastel and Remenik (2019 (arXiv:1908.10353)) found a remarkable relation between some solutions of the finite time Kardar–Parisi–Zhang (KPZ) equation and the Kadomtsev–Petviashvili (KP) equation. Using this relation we obtain the large deviations at large time and at short time for the KPZ equation with droplet initial conditions, and at a short time with half-Brownian initial conditions. It is consistent with previous results and allows us to obtain sub-leading corrections, as well as results at intermediate time. In addition, we find that the appropriate generating function associated to the full Brownian initial condition also satisfies the KP equation. Finally, generating functions for some linear statistics of the Airy point process are also found to satisfy the KP property, and consequences are discussed.

Large Gap Asymptotics for Airy Kernel Determinants with Discontinuities

We obtain large gap asymptotics for Airy kernel Fredholm determinants with any number m of discontinuities. These m-point determinants are generating functions for the Airy point process and encode probabilistic information about eigenvalues near soft edges in random matrix ensembles. Our main result is that the m-point determinants can be expressed asymptotically as the product of m 1-point determinants, multiplied by an explicit constant pre-factor which can be interpreted in terms of the covariance of the counting function of the process.

Systematic time expansion for the Kardar–Parisi–Zhang equation, linear statistics of the GUE at the edge and trapped fermions

We present a systematic short time expansion for the generating function of the one point height probability distribution for the KPZ equation with droplet initial condition, which goes much beyond previous studies. The expansion is checked against a numerical evaluation of the known exact Fredholm determinant expression. We also obtain the next order term for the Brownian initial condition. Although initially devised for short time, a resummation of the series allows to obtain also the long time large deviation function, found to agree with previous works using completely different techniques. Unexpected similarities with stationary large deviations of TASEP with periodic and open boundaries are discussed. Two additional applications are given. (i) Our method is generalized to study the linear statistics of the Airy point process, i.e. of the GUE edge eigenvalues. We obtain the generating function of the cumulants of the empirical measure to a high order. The second cumulant is found to match the result in the bulk obtained from the Gaussian free field by Borodin and Ferrari [1,2], but we obtain systematic corrections to the Gaussian free field (higher cumulants, expansion towards the edge). This also extends a result of Basor and Widom [3] to a much higher order. We obtain large deviation functions for the Airy point process for a variety of linear statistics test functions. (ii) We obtain results for the counting statistics of trapped fermions at the edge of the Fermi gas in both the high and the low temperature limits.

Coulomb-Gas Electrostatics Controls Large Fluctuations of the Kardar-Parisi-Zhang Equation.

We establish a large deviation principle for the Kardar-Parisi-Zhang (KPZ) equation, providing precise control over the left tail of the height distribution for narrow wedge initial condition. Our analysis exploits an exact connection between the KPZ one-point distribution and the Airy point process-an infinite particle Coulomb gas that arises at the spectral edge in random matrix theory. We develop the large deviation principle for the Airy point process and use it to compute, in a straightforward and assumption-free manner, the KPZ large deviation rate function in terms of an electrostatic problem (whose solution we evaluate). This method also applies to the half-space KPZ equation, showing that its rate function is half of the full-space rate function. In addition to these long-time estimates, we provide rigorous proof of finite-time tail bounds on the KPZ distribution, which demonstrate a crossover between exponential decay with exponent 3 (in the shallow left tail) to exponent 5/2 (in the deep left tail). The full-space KPZ rate function agrees with the one computed in Sasorov etal. [J. Stat. Mech. (2017) 063203JSMTC61742-546810.1088/1742-5468/aa73f8] via a WKB approximation analysis of a nonlocal, nonlinear integrodifferential equation generalizing Painlevé II which Amir etal. [Commun. Pure Appl. Math. 64, 466 (2011)CPMAMV0010-364010.1002/cpa.20347] related to the KPZ one-point distribution.

The Generating Function for the Airy Point Process and a System of Coupled Painlevé II Equations

AbstractFor 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 kth 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 kth and the ℓth largest eigenvalue or the sum of the k largest eigenvalues, can be expressed in terms of Fredholm determinants of an Airy kernel with several discontinuities. We establish simple Tracy–Widom type expressions for these Fredholm determinants, which involve solutions to systems of coupled Painlevé II equations, and we investigate the asymptotic behavior of these solutions.

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 correlations visible at the liquid-gas boundary. We prove that this random measure converges to the extended Airy point process. This indicates that, in a sense, the liquid-gas boundary should also be described by the Airy process.

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.

Asymptotic domino statistics in the Aztec diamond

We study random domino tilings of the Aztec diamond with different weights for horizontal and vertical dominoes. A domino tiling of an Aztec diamond can also be described by a particle system which is a determinantal process. We give a relation between the correlation kernel for this process and the inverse Kasteleyn matrix of the Aztec diamond. This gives a formula for the inverse Kasteleyn matrix which generalizes a result of Helfgott. As an application, we investigate the asymptotics of the process formed by the southern dominoes close to the frozen boundary. We find that at the northern boundary, the southern domino process converges to a thinned Airy point process. At the southern boundary, the process of holes of the southern domino process converges to a multiple point process that we call the thickened Airy point process. We also study the convergence of the domino process in the unfrozen region to the limiting Gibbs measure.

A Limit Theorem at the Spectral Edge for Corners of Time-Dependent Wigner Matrices

For the eigenvalues of principal submatrices of stochastically evolving Wigner matrices, we construct and study the edge scaling limit: a random decreasing sequence of continuous functions of two variables, which at every point has the distribution of the Airy point process. The analysis is based on the methods developed by Soshnikov to study the extreme eigenvalues of a single Wigner matrix.

The spectral edge of some random band matrices

We study the asymptotic distribution of the eigenvalues of random Hermitian periodic band matrices, focusing on the spectral edges. The eigenvalues close to the edges converge in distribution to the Airy point process if (and only if) the band is sufficiently wide (W ≫ N 5/6 ). Otherwise, a different limiting distribution appears.

Edge scaling limits for a family of non-Hermitian random matrix ensembles

A family of random matrix ensembles interpolating between the GUE and the Ginibre ensemble of $n\times n$ matrices with iid centered complex Gaussian entries is considered. The asymptotic spectral distribution in these models is uniform in an ellipse in the complex plane, which collapses to an interval of the real line as the degree of non-Hermiticity diminishes. Scaling limit theorems are proven for the eigenvalue point process at the rightmost edge of the spectrum, and it is shown that a non-trivial transition occurs between Poisson and Airy point process statistics when the ratio of the axes of the supporting ellipse is of order $n^{-1/3}$. In this regime, the family of limiting probability distributions of the maximum of the real parts of the eigenvalues interpolates between the Gumbel and Tracy-Widom distributions.