We study a family of distributions that arise in critical unitary random matrix ensembles. They are expressed as Fredholm determinants and describe the limiting distribution of the largest eigenvalue when the dimension of the random matrices tends to infinity. The family contains the Tracy-Widom distribution and higher order analogues of it. We compute the distributions numerically by solving a Riemann-Hilbert problem numerically, plot the distributions, and discuss several properties that they appear to exhibit.
Using loop equations, we compute the large deviation function of the maximum eigenvalue to the right of the spectrum in the Gaussian beta matrix ensembles, to all orders in 1/N. We then give a physical derivation of the all order asymptotic expansion of the right tail Tracy-Widom beta laws, for all positive beta, by studying the double scaling limit.
The Tracy-Widom distribution that has been much studied in recent years can be thought of as an extreme value distribution. We discuss interpolation between the classical extreme value distribution $\exp(-\exp(-x))$, the Gumbel distribution and the Tracy-Widom distribution. There is a family of determinantal processes whose edge behaviour interpolates between a Poisson process with density $\exp(-x)$ and the Airy kernel point process. This process can be obtained as a scaling limit of a grand canonical version of a random matrix model introduced by Moshe, Neuberger and Shapiro. We also conside…
The Tracy-Widom distribution functions involve integrals of a Painlevé II function starting from positive infinity. In this paper, we express the Tracy-Widom distribution functions in terms of integrals starting from minus infinity. There are two consequences of these new representations. The first is the evaluation of the total integral of the Hastings-McLeod solution of the Painlevé II equation. The second is the evaluation of the constant term of the asymptotic expansions of the Tracy-Widom distribution functions as the distribution parameter approaches minus infinity. For the GUE Tracy-Wid…
The distribution function of the free energy fluctuations in one-dimensional directed polymers with free boundary conditions is derived by mapping the replicated problem to the N-particle quantum boson system with attractive interactions. It is shown that in the thermodynamic limit this function is described by the universal Tracy-Widom distribution of the Gaussian orthogonal ensemble.
Investigating the long time asymptotics of the totally asymmetric simple exclusion process, Sasamoto obtains rather indirectly a formula for the GOE Tracy-Widom distribution. We establish that his novel formula indeed agrees with more standard expressions.
While originally discovered in the context of the Gaussian Unitary Ensemble, the Tracy-Widom distribution also rules the height fluctuations of growth processes. This suggests that there might be other nonequilibrium processes in which the Tracy-Widom distribution plays an important role. In our contribution we study one-dimensional systems with domain wall initial conditions. For an appropriate choice of parameters the profile develops a rarefaction wave, while maintaining the initial equilibrium states far to the left and right, which thus serve as infinitely extended thermal reservoirs. For…
We consider TASEP in continuous time with non-random initial conditions and arbitrary fixed density of particles rho. We show GOE Tracy-Widom universality of the one-point fluctuations of the associated height function. The result phrased in last passage percolation language is the universality for the point-to-line problem where the line has an arbitrary slope.
We consider spectral properties and the edge universality of sparse random matrices, the class of random matrices that includes the adjacency matrices of the Erdos-Renyi graph model $G(N,p)$. We prove a local law for the eigenvalue density up to the spectral edges. Under a suitable condition on the sparsity, we also prove that the rescaled extremal eigenvalues exhibit GOE Tracy-Widom fluctuations if a deterministic shift of the spectral edge due to the sparsity is included. For the adjacency matrix of the Erdos-Renyi graph this establishes the Tracy-Widom fluctuations of the second largest eig…
Consider the random matrix obtained from the adjacency matrix of a random d-regular graph by multiplying every entry by a random sign. The largest eigenvalue converges, after proper scaling, to the Tracy--Widom distribution.
We study Fredholm determinants of the Painlevé II and Painlevé XXXIV kernels. In certain critical unitary random matrix ensembles, these determinants describe special gap probabilities of eigenvalues. We obtain Tracy-Widom formulas for the Fredholm determinants, which are explicitly given in terms of integrals involving a family of distinguished solutions to the coupled Painlevé II system in dimension four. Moreover, the large gap asymptotics for these Fredholm determinants are derived, where the constant terms are given explicitly in terms of the Riemann zeta-function.
We analyze the left-tail asymptotics of deformed Tracy-Widom distribution functions describing the fluctuations of the largest eigenvalue in invariant random matrix ensembles after removing each soft edge eigenvalue independently with probability $1-γ\in[0,1]$. As $γ$ varies, a transition from Tracy-Widom statistics ($γ=1$) to classical Weibull statistics ($γ=0$) was observed in the physics literature by Bohigas, de Carvalho, and Pato \cite{BohigasCP:2009}. We provide a description of this transition by rigorously computing the leading-order left-tail asymptotics of the thinned GOE, GUE and GS…
We study Fredholm determinants related to a family of kernels which describe the edge eigenvalue behavior in unitary random matrix models with critical edge points. The kernels are natural higher order analogues of the Airy kernel and are built out of functions associated with the Painlevé I hierarchy. The Fredholm determinants related to those kernels are higher order generalizations of the Tracy-Widom distribution. We give an explicit expression for the determinants in terms of a distinguished smooth solution to the Painlevé II hierarchy. In addition we compute large gap asymptotics for the …
We establish the relation between two objects: an integrable system related to Painleve II equation, and the symplectic invariants of a certain plane curve Σ_{TW} describing the average eigenvalue density of a random hermitian matrix spectrum near a hard edge (a bound for its maximal eigenvalue). This explains directly how the Tracy-Widow law F_{GUE}, governing the distribution of the maximal eigenvalue in hermitian random matrices, can also be recovered from symplectic invariants.
We study the growing patterns formed by a deterministic cellular automaton, the rotor-router model, in the presence of quenched noise. By the detailed study of two cases, we show that: (a) the boundary of the pattern displays KPZ fluctuations with a Tracy-Widom distribution, (b) as one increases the amount of randomness, the rotor-router path undergoes a transition from a recurrent to a transient walk. This transition is analysed here for the first time, and it is shown that it falls in the 3D Anisotropic Directed Percolation universality class.
We show that the fluctuations of the largest eigenvalue of any generalized Wigner matrix $H$ converge to the Tracy-Widom laws at a rate nearly $O(N^{-1/3})$, as the matrix dimension $N$ tends to infinity. We allow the variances of the entries of $H$ to have distinct values but of comparable sizes such that $\sum_{i} \mathbb{E}|h_{ij}|^2=1$. Our result improves the previous rate $O(N^{-2/9})$ by Bourgade [8] and the proof relies on the first long-time Green function comparison theorem near the edges without the second moment matching restriction.
We study an oriented first passage percolation model for the evolution of a river delta. This model is exactly solvable and occurs as the low temperature limit of the beta random walk in random environment. We analyze the asymptotics of an exact formula from [4] to show that, at any fixed positive time, the width of a river delta of length $L$ approaches a constant times $L^{2/3}$ with Tracy-Widom GUE fluctuations of order $L^{4/9}$. This result can be rephrased in terms of particle systems. We introduce an exactly solvable particle system on the integer half line and show that after running t…
This is a survey of Harold Widom's work in random matrices. We start with his pioneering papers on the sine-kernel determinant, continue with his and Craig Tracy's groundbreaking results concerning the distribution functions of random matrix theory, touch on the remarkable universality of the Tracy-Widom distributions in mathematics and physics, and close with Tracy and Widom's remarkable work on the asymmetric simple exclusion process.
Two approaches (TW and ASvM) to derivation of integrable differential equations for random matrix probabilities are compared. Both methods are rewritten in such a form that simple and explicit relations between all TW dependent variables and $τ$-functions of ASvM are found, for the example of finite size Gaussian matrices. Orthogonal function systems and Toda lattice are seen as the core structure of both approaches and their relationship.
We construct a random Schrodinger operator on a subset of the hexagonal lattice and study its smallest positive eigenvalues. Using an asymptotic mapping, we relate them to the partition function of the directed polymer model on the square lattice. For a specific choice of the edge weight distribution, we obtain a model known as the log-Gamma polymer, which is integrable. Recent results about the fluctuations of free energy for the log-Gamma polymer allow us to prove Tracy-Widom type fluctuations for the smallest eigenvalue of the random Schrodinger operator. We also relate the distribution of …
We study the Tracy-Widom distribution function for Dyson's $β$-ensemble with $β= 6$. The starting point of our analysis is the recent work of I. Rumanov where he produces a Lax-pair representation for the Bloemendal-Virág equation. The latter is a linear PDE which describes the Tracy-Widom functions corresponding to general values of $β$. Using his Lax pair, Rumanov derives an explicit formula for the Tracy-Widom $β=6$ function in terms of the second Painlevé transcendent and the solution of an auxiliary ODE. Rumanov also shows that this formula allows him to derive formally the asymptotic exp…
We show that the fluctuations of the largest eigenvalue of a real symmetric or complex Hermitian Wigner matrix of size $N$ converge to the Tracy--Widom laws at a rate $O(N^{-1/3+ω})$, as $N$ tends to infinity. For Wigner matrices this improves the previous rate $O(N^{-2/9+ω})$ obtained by Bourgade [5] for generalized Wigner matrices. Our result follows from a Green function comparison theorem, originally introduced by Erdős, Yau and Yin [19] to prove edge universality, on a finer spectral parameter scale with improved error estimates. The proof relies on the continuous Green function flow indu…
We determine completely the Tracy-Widom distribution for Dyson's β-ensemble with β=6. The problem of the Tracy-Widom distribution of β-ensemble for general β>0 has been reduced to find out a bounded solution of the Bloemendal-Virág equation with a specified boundary. Rumanov proposed a Lax pair approach to solve the Bloemendal-Virág equation for even integer β. He also specially studied the β=6 case with his approach and found a second order nonlinear ordinary differential equation (ODE) for the logarithmic derivative of the Tracy-Widom distribution for \bea=6. Grava et al. continued to study …
In this paper, we first briefly review some recent results on the distribution of the maximal eigenvalue of a $(N\times N)$ random matrix drawn from Gaussian ensembles. Next we focus on the Gaussian Unitary Ensemble (GUE) and by suitably adapting a method of orthogonal polynomials developed by Gross and Matytsin in the context of Yang-Mills theory in two dimensions, we provide a rather simple derivation of the Tracy-Widom law for GUE. Our derivation is based on the elementary asymptotic scaling analysis of a pair of coupled nonlinear recursion relations. As an added bonus, this method also all…
We study the distribution of the largest eigenvalue in formal Hermitian one-matrix models at multicriticality, where the spectral density acquires an extra number of k-1 zeros at the edge. The distributions are directly expressed through the norms of orthogonal polynomials on a semi-infinite interval, as an alternative to using Fredholm determinants. They satisfy non-linear recurrence relations which we show form a Lax pair, making contact to the string literature in the early 1990's. The technique of pseudo-differential operators allows us to give compact expressions for the logarithm of the …
We exploit a relation between the mean number $N_{m}$ of minima of random Gaussian surfaces and extreme eigenvalues of random matrices to understand the critical behaviour of $N_{m}$ in the simplest glass-like transition occuring in a toy model of a single particle in $N$-dimensional random environment, with $N\gg 1$. Varying the control parameter $μ$ through the critical value $μ_c$ we analyse in detail how $N_{m}(μ)$ drops from being exponentially large in the glassy phase to $N_{m}(μ)\sim 1$ on the other side of the transition. We also extract a subleading behaviour of $N_{m}(μ)$ in both gl…
In arXiv:1306.2117, we found explicit Lax pairs for the soft edge of beta ensembles with even integer values of $β$. Using this general result, the case $β=6$ is further considered here. This is the smallest even $β$, when the corresponding Lax pair and its relation to Painlevé II (PII) have not been known before, unlike cases $β=2$ and $4$. It turns out that again everything can be expressed in terms of the Hastings-McLeod solution of PII. In particular, a second order nonlinear ODE for the logarithmic derivative of Tracy-Widom distribution for $β=6$ involving the PII function in the coeffici…
We study the distribution of the largest eigenvalue in the "Pfaffian" classical ensembles of random matrix theory, namely in the Gaussian orthogonal (GOE) and Gaussian symplectic (GSE) ensembles, using semi-classical skew-orthogonal polynomials, in analogue to the approach of Nadal and Majumdar (NM) for the Gaussian unitary ensemble (GUE). Generalizing the techniques of Adler, Forrester, Nagao and van Moerbeke, and using "overlapping Pfaffian" identities due to Knuth, we explicitly construct these semi-classical skew-orthogonal polynomials in terms of the semi-classical orthogonal polynomials …
We present an analytic expression of the nonperturbative free energy of a double-well supersymmetric matrix model in its double scaling limit, which corresponds to two-dimensional type IIA superstring theory on a nontrivial Ramond-Ramond background. To this end we draw upon the wisdom of random matrix theory developed by Tracy and Widom, which expresses the largest eigenvalue distribution of unitary ensembles in terms of a Painleve II transcendent. Regularity of the result at any value of the string coupling constant shows that the third-order phase transition between a supersymmetry-preservin…
The Jánossy density for a determinantal point process is the probability density that an interval $I$ contains exactly $p$ points except for those at $k$ designated loci. The Jánossy density associated with an integrable kernel $\mathbf{K}\doteq (φ(x)ψ(y)-ψ(x)φ(y))/(x-y)$ is shown to be expressed as a Fredholm determinant $\mathrm{Det}(\mathbb{I}-\tilde{\mathbf{K}}|_I)$ of a transformed kernel $\tilde{\mathbf{K}}\doteq (\tildeφ(x)\tildeψ(y)-\tildeψ(x)\tildeφ(y))/(x-y)$. We observe that $\tilde{\mathbf{K}}$ satisfies Tracy and Widom's criteria if $\mathbf{K}$ does, because of the structure that…
We study a system of N non-intersecting Brownian motions on a line segment [0,L] with periodic, absorbing and reflecting boundary conditions. We show that the normalized reunion probabilities of these Brownian motions in the three models can be mapped to the partition function of two-dimensional continuum Yang-Mills theory on a sphere respectively with gauge groups U(N), Sp(2N) and SO(2N). Consequently, we show that in each of these Brownian motion models, as one varies the system size L, a third order phase transition occurs at a critical value L=L_c(N)\sim \sqrt{N} in the large N limit. Clos…
We study discrete $β$-ensembles as introduced in [17]. We obtain rigidity estimates on the particle locations, i.e. with high probability, the particles are close to their classical locations with an optimal error estimate. We prove the edge universality of the discrete $β$-ensembles, i.e. for $β\geq 1$, the distribution of extreme particles converges to the Tracy-Widom $β$ distribution. As far as we know, this is the first proof of general Tracy-Widom $β$ distributions in the discrete setting. A special case of our main results implies that under the Jack deformation of the Plancherel measure…
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…
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/…
Using the method of Tracy and Widom we rederive the correlation functions for β=1 Hermitian and real asymmetric ensembles of N x N matrices with N odd.
We determined the probability distribution of the combined output power from twenty five coupled fiber lasers and show that it agrees well with the Tracy-Widom, Majumdar-Vergassola and Vivo-Majumdar-Bohigas distributions of the largest eigenvalue of Wishart random matrices with no fitting parameters. This was achieved with $500,000$ measurements of the combined output power from the fiber lasers, that continuously changes with variations of the fiber lasers lengths. We show experimentally that for small deviations of the combined output power over its mean value the Tracy-Widom distribution is…
We study large random partitions boxed into a rectangle and coming from skew Howe duality, or alternatively from dual Schur measures. As the sides of the rectangle go to infinity, we obtain: 1) limit shape results for the profiles generalizing the Vershik--Kerov--Logan--Shepp curve; and 2) universal edge asymptotic results for the first parts in the form of the Tracy--Widom distribution, as well as less-universal critical regime results introduced by Gravner, Tracy and Widom. We do this for a large class of Schur parameters going beyond the Plancherel or principal specializations previously st…
This paper studies the mixing behavior of the Asymmetric Simple Exclusion Process (ASEP) on a segment of length $N$. Our main result is that for particle densities in $(0,1),$ the total-variation cutoff window of ASEP is $N^{1/3}$ and the cutoff profile is $1-F_{\mathrm{GUE}},$ where $F_{\mathrm{GUE}}$ is the Tracy-Widom distribution function. This also gives a new proof of the cutoff itself, shown earlier by Labbé and Lacoin. Our proof combines coupling arguments, the result of Tracy-Widom about fluctuations of ASEP started from the step initial condition, and exact algebraic identities comin…
In this paper, we prove a necessary and sufficient condition for Tracy-Widom law of Wigner matrices. Consider $N \times N$ symmetric Wigner matrices $H$ with $H_{ij} = N^{-1/2} x_{ij}$, whose upper right entries $x_{ij}$ $(1\le i< j\le N)$ are $i.i.d.$ random variables with distribution $μ$ and diagonal entries $x_{ii}$ $(1\le i\le N)$ are $i.i.d.$ random variables with distribution $\wt μ$. The means of $μ$ and $\wt μ$ are zero, the variance of $μ$ is 1, and the variance of $\wt μ$ is finite. We prove that Tracy-Widom law holds if and only if $\lim_{s\to \infty}s^4\p(|x_{12}| \ge s)=0$. The s…
We prove that the largest eigenvalues of the beta ensembles of random matrix theory converge in distribution to the low-lying eigenvalues of the random Schroedinger operator -d^2/dx^2 + x + (2/beta^{1/2}) b_x' restricted to the positive half-line, where b_x' is white noise. In doing so we extend the definition of the Tracy-Widom(beta) distributions to all beta>0, and also analyze their tails. Last, in a parallel development, we provide a second characterization of these laws in terms of a one-dimensional diffusion. The proofs rely on the associated tridiagonal matrix models and a universality …
We investigate the probability distribution of the length of the second row of a Young diagram of size $N$ equipped with Plancherel measure. We obtain an expression for the generating function of the distribution in terms of a derivative of an associated Fredholm determinant, which can then be used to show that as $N\to\infty$ the distribution converges to the Tracy-Widom distribution [TW] for the second largest eigenvalue of a random GUE matrix. This paper is a sequel to [BDJ], where we showed that as $N\to\infty$ the distribution of the length of the first row of a Young diagram, or equivale…
Recently Johansson and Johnstone proved that the distribution of the (properly rescaled) largest principal component of the complex (real) Wishart matrix $ X^* \* X (X^t \*X) $ converges to the Tracy-Widom law as $ n, p $ (the dimensions of $ X $) tend to $ \infty $ in some ratio $ n/p \to γ>0. $ We extend these results in two directions. First of all, we prove that the joint distribution of the first, second, third, etc. eigenvalues of a Wishart matrix converges (after a proper rescaling) to the Tracy-Widom distribution. Second of all, we explain how the combinatorial machinery developed for …
We study the edge behavior of inhomogeneous one-dimensional quantum systems, such as Lieb-Liniger models in traps or spin chains in spatially varying magnetic fields. For free systems these fall into several universality classes, the most generic one being governed by the Tracy-Widom distribution. We investigate in this paper the effect of interactions. Using semiclassical arguments, we show that since the density vanishes to leading order, the strong interactions in the bulk are renormalized to zero at the edge, which simply explains the survival of Tracy-Widom scaling in general. For integra…
Directed Polymer Transfer Matrices as a Unified Generator of Distinct One-Point Fluctuation Laws
Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）；School of Science, Constructor University（Constructor大学理学院）；Department of Physics, Massachusetts Institute of Technology（麻省理工学院物理系）
We consider the solution of the stochastic heat equation \partial_T \mathcal{Z} = 1/2 \partial_X^2 \mathcal{Z} - \mathcal{Z} \dot{\mathscr{W}} with delta function initial condition \mathcal{Z} (T=0)= δ_0 whose logarithm, with appropriate normalizations, is the free energy of the continuum directed polymer, or the solution of the Kardar-Parisi-Zhang equation with narrow wedge initial conditions. We obtain explicit formulas for the one-dimensional marginal distributions -- the {\it crossover distributions} -- which interpolate between a standard Gaussian distribution (small time) and the GUE Tra…
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.
We introduce an exactly-solvable model of random walk in random environment that we call the Beta RWRE. This is a random walk in $\mathbb{Z}$ which performs nearest neighbour jumps with transition probabilities drawn according to the Beta distribution. We also describe a related directed polymer model, which is a limit of the $q$-Hahn interacting particle system. Using a Fredholm determinant representation for the quenched probability distribution function of the walker's position, we are able to prove second order cube-root scale corrections to the large deviation principle satisfied by the w…
We present a random matrix interpretation of the distribution functions which have appeared in the study of the one-dimensional polynuclear growth (PNG) model with external sources. It is shown that the distribution, GOE$^2$, which is defined as the square of the GOE Tracy-Widom distribution, can be obtained as the scaled largest eigenvalue distribution of a special case of a random matrix model with a deterministic source, which have been studied in a different context previously. Compared to the original interpretation of the GOE$^2$ as ``the square of GOE'', ours has an advantage that it ca…
Tracy and Widom showed that the level spacing function of the Gaussian unitary ensemble is related to a particular solution of the fourth Painlevé equation. We reconsider this problem from the viewpoint of Hirota's bilinear method in soliton theory and present another proof. We also consider the asymptotic behavior of the level spacing function as $s\to\infty$, and its relation to the "Clarkson-McLeod solution" to the Painlevé IV equation.
In these lecture we explain why limiting distribution function, like the Tracy-Widom distribution, or limit processes, like the Airy_2 process, arise both in random matrices and interacting particle systems. The link is through a common mathematical structure on an interlacing structure, also known as Gelfand-Tsetlin pattern, that appears for specific models in both fields.
