We investigate the characteristic polynomials $φ_N$ of the Gaussian $β$-ensemble for general $β>0$ through its transfer matrix recurrence. Our motivation is to obtain a (probabilistic) approximation for $φ_N$ in terms of a Gaussian log--correlated field in order to ultimately deduce some of its fine asymptotic properties. We distinguish between different types of transfer matrices and analyze completely the hyperbolic regime of the recurrence. As a result, we obtain a new coupling between $φ_N(z)$ and a Gaussian analytic function with an error which is uniform for $z \in \mathbb{C}$ separated from the support of the semicircle law. We use this as input to give the almost sure scaling limit of the characteristic polynomial at the edge in arXiv:2009.05003. This is also required to obtain analogous strong approximations inside of the bulk of the semicircle law. Our analysis relies on moderate deviation estimates for the product of transfer matrices and this approach might also be useful in different contexts.
Let $λ$ and $μ$ be beta-Jacobi and beta-Laguerre ensembles with joint density function $f_{β, m, a_1, a_2}$ and $f_{β, m, a_1}$, respectively. Here $β>0$ and $a_1, a_2$ and $m$ satisfying . $a_1, a_2>\fracβ{2}(m-1).$ In this paper, we consider the distance between $2(a_1+a_2)λ$ and $μ$ in terms of total variation distance and Kullback-Leibler distance. Following the idea in \cite{JM2017}, we are able to prove that both the two distances go to zero once $a_1m=o(a_2)$ and not so if $\lim_{a_2\to\infty}a_1m/a_2=σ>0.$
We establish various small deviation inequalities for the extremal (soft edge) eigenvalues in the beta-Hermite and beta-Laguerre ensembles. In both settings, upper bounds on the variance of the largest eigenvalue of the anticipated order follow immediately.
This paper proves a matrix model for the Wishart Ensemble with general covariance and general dimension parameter beta. In so doing, we introduce a new and elegant definition of Jack polynomials.
We provide a precise coupling of the finite circular beta ensembles and their limit process via their operator representations. We prove explicit bounds on the distance of the operators and the corresponding point processes. We also prove an estimate on the beta-dependence in the $\text{Sine}_β$ process.
We consider generalized Wigner ensembles and general beta-ensembles with analytic potentials for any beta larger than 1. The recent universality results in particular assert that the local averages of consecutive eigenvalue gaps in the bulk of the spectrum are universal in the sense that they coincide with those of the corresponding Gaussian beta-ensembles. In this article, we show that local averaging is not necessary for this result, i.e. we prove that the single gap distributions in the bulk are universal.
As an important topic in Mathematical Physics and statistics, random matrices theory has found uses in many aspects of modern physics and multivariate analysis. This paper is to investigate the Gaussian fluctuations for linear spectral statistics (LSS) of Wigner beta ensembles. We first establish a central limit theorem (CLT) for LSS of Wigner quaternion matrices, then give a general CLT for Wigner $β$ ensembles.
The bead process introduced by Boutillier is a countable interlacing of the determinantal sine-kernel point processes. We construct the bead process for general sine beta processes as an infinite dimensional Markov chain whose transition mechanism is explicitly described. We show that this process is the microscopic scaling limit in the bulk of the Hermite beta corner process introduced by Gorin and Shkolnikov, generalizing the process of the minors of the Gaussian unitary and orthogonal ensembles. In order to prove our results, we use bounds on the variance of the point counting of the circul…
The paper describes the global limiting behavior of Gaussian beta ensembles where the parameter $β$ is allowed to vary with the matrix size $n$. In particular, we show that as $n \to \infty$ with $nβ\to \infty$, the empirical distribution converges weakly to the semicircle distribution, almost surely. The Gaussian fluctuation around the limit is then derived by a martingale approach.
For beta ensembles with convex poynomial potentials, we prove a large deviation principle for the empirical spectral distribution seen from the rightmost particle. This modified spectral distribution was introduced by Perret and Schehr (J. Stat. Phys. 2014) to study the crowding near the maximal eigenvalue, in the case of the GUE. We prove also convergence of fluctuations.
We study the limiting behavior of Gaussian beta ensembles in the regime where $βn = const$ as $n \to \infty$. The results are (1) Gaussian fluctuations for linear statistics of the eigenvalues, and (2) Poisson convergence of the bulk statistics. (2) is an alternative proof of the result by F.~Benaych-Georges and S.~Péché (2015) with the explicit form of the intensity measure.
We consider the Gaussian beta-ensemble when $β$ scales with $n$ the number of particles such that $\displaystyle{{n}^{-1}\ll β\ll 1}$. Under a certain regime for $β$, we show that the largest particle satisfies a large deviations principle in $\mathbb{R}$ with speed $nβ$ and explicit rate function. As a consequence, the largest particle converges in probability to $2$, the rightmost point of the semicircle law.
For Y a subset of the complex plane,a beta ensemble is a sequence of probability measures on Y^n for n=1,2,3...depending on a real-valued continuous function Q and a real positive parameter beta.We consider the associated sequence of probability measures on Y where the probability of a subset W is given by the probability that at least one coordinate of Y^n belongs to W. With appropriate restrictions on Y,Q we prove a large deviation principle for this sequence of measures. This extends a result of Borot-Guionnet to subsets of the complex plane and to beta ensembles defined with measures using…
In this note, we prove that the normalized trace of the resolvent of the beta-Laguerre ensemble eigenvalues is close to the Stieltjes transform of the Marchenko-Pastur (MP) distribution with very high probability, for values of the imaginary part greater than m^{-1+ε}. As an immediate corollary, we obtain convergence of the one-point density to the MP law on short scales. The proof serves to illustrate some simplifications of the method introduced in our previous work to prove a local semi-circle law for Gaussian beta-ensembles.
Beta ensembles on the real line with three classical weights (Gaussian, Laguerre and Jacobi) are now realized as the eigenvalues of certain tridiagonal random matrices. The paper deals with beta Jacobi ensembles, the type with the Jacobi weight. Making use of the random matrix model, we show that in the regime where $βN \to const \in [0, \infty)$, with $N$ the system size, the empirical distribution of the eigenvalues converges weakly to a limiting measure which belongs to a new class of probability measures of associated Jacobi polynomials. This is analogous to the existing results for the ot…
We consider the beta-Laguerre ensemble, a family of distributions generalizing the joint eigenvalue distribution of the Wishart random matrices. We show that the bulk scaling limit of these ensembles exists for all beta>0 for a general family of parameters and it is the same as the bulk scaling limit of the corresponding beta-Hermite ensemble.
In this paper we show weak convergence of the empirical eigenvalue distribution and of the weighted spectral measure of the Jacobi ensemble, when one or both parameters grow faster than the dimension $n$. In these cases the limit measure is given by the Marchenko-Pastur law and the semicircle law, respectively. For the weighted spectral measure we also prove large deviation principles under this scaling, where the rate functions are those of the other classical ensembles.
Beta Laguerre ensembles which are generalizations of Wishart ensembles and Laguerre ensembles can be realized as eigenvalues of certain random tridiagonal matrices. Analogous to the Wishart ($β=1$) case and the Laguerre ($β= 2$) case, for fixed $β$, it is known that the empirical distribution of the eigenvalues of these ensembles converges weakly to Marchenko--Pastur distributions, almost surely. The paper restudies the limiting behavior of the empirical distribution but in regimes where the parameter $β$ is allowed to vary as a function of the matrix size $N$. We show that the above Marchenko…
We give a simple proof of a central limit theorem for linear statistics of the Circular beta-ensembles which is valid at almost arbitrary mesoscopic scale and for functions of class C^3. As a consequence, using a coupling introduced by Valko and Virag, we deduce a central limit theorem for the Sine beta processes. We also discuss the connection between our result and the theory of Gaussian Multiplicative Chaos. Based on the results of Lambert-Ostrovsky-Simm, we show that the exponential of the logarithm of the real (and imaginary) part of the characteristic polynomial of the Circular beta-ense…
This paper studies beta ensembles on the real line in a high temperature regime, that is, the regime where $βN \to const \in (0, \infty)$, with $N$ the system size and $β$ the inverse temperature. In this regime, the convergence to the equilibrium measure is a consequence of a recent result on large deviation principle by Liu and Wu (Stochastic Processes and their Applications (2019)). This paper focuses on the local behavior and shows that the local statistics around any fixed reference energy converges weakly to a homogeneous Poisson point process.
Motivated by problems of hyperbolic stochastic geometry we introduce and study the class of beta-star polytopes. A beta-star polytope is defined as the convex hull of an inhomogeneous Poisson processes on the complement of the unit ball in $\mathbb{R}^d$ with density proportional to $(||x||^2-1)^β$, where $||x|| > 1$ and $β>d/2$. Explicit formulas for various geometric and combinatorial functionals associated with beta-star polytopes are provided, including the expected number of $k$-dimensional faces, the expected external angle sums and the expected intrinsic volumes. Beta-star polytopes are…
We investigate the random eigenvalues coming from the beta-Laguerre ensemble with parameter p, which is a generalization of the real, complex and quaternion Wishart matrices of parameter (n,p). In the case that the sample size n is much smaller than the dimension of the population distribution p, a common situation in modern data, we approximate the beta-Laguerre ensemble by a beta-Hermite ensemble which is a generalization of the real, complex and quaternion Wigner matrices. As corollaries, when n is much smaller than p, we show that the largest and smallest eigenvalues of the complex Wishart…
Beta Ensembles in the Freezing Regime and Finite Free Convolutions
Mathematical Institute, Tohoku University（东北大学数学研究所）；Global Center for Science and Engineering, Waseda University（早稻田大学全球科学与工程中心）；Graduate School of Fundamental Science and Engineering, Waseda University（早稻田大学理工学术院）
We study the asymptotic edge statistics of the Gaussian $β$-ensemble, a collection of $n$ particles, as the inverse temperature $β$ tends to zero as $n$ tends to infinity. In a certain decay regime of $β$, the associated extreme point process is proved to converge in distribution to a Poisson point process as $n\to +\infty$. We also extend a well known result on Poisson limit for Gaussian extremes by showing the existence of an edge regime that we did not find in the literature.
Bourgade, Nikeghbali and Rouault recently proposed a matrix model for the circular Jacobi $β$-ensemble, which is a generalization of the Dyson circular $β$-ensemble but equipped with an additional parameter $b$, and further studied its limiting spectral measure. We calculate the scaling limits for expected products of characteristic polynomials of circular Jacobi $β$-ensembles. For the fixed constant $b$, the resulting limit near the spectrum singularity is proven to be a new multivariate function. When $b=βNd/2$, the scaling limits in the bulk and at the soft edge agree with those of the Herm…
We study the dimension properties of the spectral measure of the Circular $β$-Ensembles. For $β\geq 2$ it it was previously shown by Simon that the spectral measure is almost surely singular continuous with respect to Lebesgue measure on $\partial \mathbb{D}$ and the dimension of its support is $1 - 2/β$. We reprove this result with a combination of probabilistic techniques and the so-called Jitomirskaya-Last inequalities. Our method is simpler in nature and mostly self-contained, with an emphasis on the probabilistic aspects rather than the analytic. We also extend the method to prove a large…
An explicit formula for the mean spectral measure of a random Jacobi matrix is derived. The matrix may be regarded as the limit of Gaussian beta ensemble (G$β$E) matrices as the matrix size $N$ tends to infinity with the constraint that $N β$ is a constant.
We show that beta ensembles in Random Matrix Theory with generic real analytic potential have the asymptotic equipartition property. In addition, we prove a Central Limit Theorem for the density of the eigenvalues of these ensembles.
We prove the edge universality of the beta ensembles for any $β\ge 1$, provided that the limiting spectrum is supported on a single interval, and the external potential is $\mathscr{C}^4$ and regular. We also prove that the edge universality holds for generalized Wigner matrices for all symmetry classes. Moreover, our results allow us to extend bulk universality for beta ensembles from analytic potentials to potentials in class $\mathscr{C}^4$.
We find the joint generalized singular value distribution and largest generalized singular value distributions of the $β$-MANOVA ensemble with positive diagonal covariance, which is general. This has been done for the continuous $β> 0$ case for identity covariance (in eigenvalue form), and by setting the covariance to $I$ in our model we get another version. For the diagonal covariance case, it has only been done for $β= 1,2,4$ cases (real, complex, and quaternion matrix entries). This is in a way the first second-order $β$-ensemble, since the sampler for the generalized singular values of the…
We give a short proof of the free analogue of the Talagrand inequality for the transportation cost to the semicircular which was originally proved by Biane and Voiculescu. The proof is based on a convexity argument and is in the spirit of the original Talagrand's proof. We also discuss the convergence, fluctuations and large deviations of the energy of the eigenvalues of beta ensembles, which gives also yet another proof of the convergence of the eigenvalue distribution to the semicircle law.
We study the averages of ratios of characteristic polynomials over circular $β$-ensembles, where $β$ is a positive real number. Using Jack polynomial theory, we obtain three expressions for ratio averages. Two of them are given as sums of super-Jack polynomials and another one is given by a hyperdeterminant. As applications, we give dual relations for ratio averages between $β$ and $4/β$.
For a beta-Jacobi ensemble determined by parameters a_1, a_2 and n, under the restriction that the three parameters go to infinity with n and a_1 being of small orders of a_2, we obtain both the bulk and the edge scaling limits. In particular, we derive the asymptotic distributions for the largest and the smallest eigenvalues, the Central Limit Theorems of the eigenvalues, and the limiting distributions of the empirical distributions of the eigenvalues.
The spectral measures of random Jacobi matrices related to beta ensembles at high temperature and Dirichlet processes
Mathematical Institute, Tohoku University（东北大学数学研究所）；Faculty of Mathematics Mechanics Informatics, University of Science, Vietnam National University（越南国立大学科学大学数学力学信息学院）；Global Center for Science and Engineering, Waseda University（早稻田大学全球科学与工程中心）
This paper constructs tridiagonal random matrix models for general ($β>0$) $β$-Hermite (Gaussian) and $β$-Laguerre (Wishart) ensembles. These generalize the well-known Gaussian and Wishart models for $β= 1,2,4$. Furthermore, in the cases of the $β$-Laguerre ensembles, we eliminate the exponent quantization present in the previously known models. We further discuss applications for the new matrix models, and present some open problems.
We study CMV matrices (a discrete one-dimensional Dirac-type operator) with random decaying coefficients. Under mild assumptions we identify the local eigenvalue statistics in the natural scaling limit. For rapidly decreasing coefficients, the eigenvalues have rigid spacing (like the numerals on a clock); in the case of slow decrease, the eigenvalues are distributed according to a Poisson process. For a certain critical rate of decay we obtain the circular beta ensembles of random matrix theory. The temperature β^{-1} appears as the square of the coupling constant.
We study limiting distribution of pair counting statistics of the form $ \sum_{1\leq i\neq j\leq N} f(L_N\*(θ_i-θ_j))$ for the circular $β$-ensemble (C$β$E) of random matrices for sufficiently smooth test function $f$ and $L_N=O(N).$ For $β=2$ and $L_N=N$ our results are inspired by a classical result of Montgomery on pair correlation of zeros of Riemann zeta function.
We introduce and study a family of $(q,t)$-deformed discrete $N$-particle beta ensembles, where $q$ and $t$ are the parameters of Macdonald polynomials. The main result is the existence of a large-$N$ limit transition leading to random point processes with infinitely many particles.
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 …
In a recent work Killip and Nenciu gave random recurrences for the characteristic polynomials of certain unitary and real orthogonal upper Hessenberg matrices. The corresponding eigenvalue p.d.f.'s are beta-generalizations of the classical groups. Left open was the direct calculation of certain Jacobians. We provide the sought direct calculation. Furthermore, we show how a multiplicative rank 1 perturbation of the unitary Hessenberg matrices provides a joint eigenvalue p.d.f generalizing the circular beta-ensemble, and we show how this joint density is related to known inter-relations between …
Bessel processes associated with the root systems $A_{N-1}$ and $B_N$ describe interacting particle systems with $N$ particles on $\mathbb R$; they form dynamic versions of the classical $β$-Hermite and Laguerre ensembles. In this paper we study corresponding Cauchy processes constructed via some subordination. This leads to $β$-Cauchy ensembles in both cases with explicit distributions. For these distributions we derive central limit theorems for fixed $N$ in the freezing regime, i.e., when the parameters tend to infinity. The results are closely related to corresponding known freezing result…
We suggest an hierarchy of all the results known so far about the connection of the asymptotics of combinatorial or representation theoretic problems with ``beta=2 ensembles'' arising in the random matrix theory. We show that all such results are, essentially, degenerations of one general situation arising from so-called generalized regular representations of the infinite symmetric group.
We consider the macroscopic large N limit of the Circular beta-Ensemble at high temperature, and its weighted version as well, in the regime where the inverse temperature scales as beta/N for some parameter beta>0. More precisely, in the large N limit, the equilibrium measure of this particle system is described as the unique minimizer of a functional which interpolates between the relative entropy (beta=0) and the weighted logarithmic energy (beta=\infty). More precisely, we provide subGaussian concentration estimates in the W1 metric for the deviations of the empirical measure to this equili…
We investigate the point process of moduli of the Ginibre and hyperbolic ensembles. We show that far from the origin and at an appropriate scale, these processes exhibit Gaussian and Poisson fluctuations. Among the possible Gaussian fluctuations, we can find white noise but also fluctuations with non-trivial covariance at a particular scale.
We access the edge of Gaussian beta ensembles with one spike by analyzing high powers of the associated tridiagonal matrix models. In the classical cases beta=1, 2, 4, this corresponds to studying the fluctuations of the largest eigenvalues of additive rank one perturbations of the GOE/GUE/GSE random matrices. In the infinite-dimensional limit, we arrive at a one-parameter family of random Feynman-Kac type semigroups, which features the stochastic Airy semigroup of Gorin and Shkolnikov [13] as an extreme case. Our analysis also provides Feynman-Kac formulas for the spiked stochastic Airy opera…
Beta-ensembles of random matrices are naturally considered as quantum integrable systems, in particular, due to their relation with conformal field theory, and more recently appeared connection with quantized Painlevé Hamiltonians. Here we demonstrate that, at least for {\it even integer} beta, these systems are classically integrable, e.g. there are Lax pairs associated with them, which we explicitly construct. To come to the result, we show that a solution of every Fokker-Planck equation in one space (and one time) dimensions can be considered as a component of an eigenvector of a Lax pair. …
