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 other two classical weights. We also study the limiting behavior of the empirical measure process of beta Jacobi processes in the same regime and obtain a dynamic version of the above.
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 $\lim_{a_2\to\infty}a_1m/a_2=σ>0.$
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.
The purpose of this paper is to study the existence and uniqueness of solutions to a system of Stochastic Differential Equations (SDEs). The coordinates are bounded by zero and one, and repulse each other according to a Coulombian like interaction force. We show the existence of strong and pathwise unique solutions to the system until the first multiple collision at zero or one, and give a sufficient condition on the parameters of the SDEs for this multiple collision not to occur in finite time.
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 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 define and study a multidimensional process that generalizes the eigenvalues of matrix Jacobi processes on the one hand and whose stationary distribution is given by the beta Jacobi ensemble on the other hand.
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 continue to explore the connections between large deviations for objects coming from random matrix theory and sum rules. This connection was established in [17] for spectral measures of classical ensembles (Gauss-Hermite, Laguerre, Jacobi) and it was extended to spectral matrix measures of the Hermite and Laguerre ensemble in [20]. In this paper, we consider the remaining case of spectral matrix measures of the Jacobi ensemble. Our main results are a large deviation principle for such measures and a sum rule for matrix measures with reference measure the Kesten-McKay law. As an important in…
We study the scaling limit of the spectrum of the β-Jacobi ensemble at the soft-edge and hard-edge for general values of β. We show that the limiting point processes correspond respectively to the stochastic Airy and Bessel point processes introduced in Ramirez-Rider-Virag and Ramirez-Rider.
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.
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 …
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…
We prove an operator level limit for the circular Jacobi $β$-ensemble. As a result, we characterize the counting function of the limit point process via coupled systems of stochastic differential equations. We also show that the normalized characteristic polynomials converge to a random analytic function, which we characterize via the joint distribution of its Taylor coefficients at zero and as the solution of a stochastic differential equation system. We also provide analogous results for the real orthogonal $β$-ensemble.
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.
It has been shown by Pittel and Romik that the random surface associated with a large rectangular Young tableau converges to a deterministic limit. We study the fluctuations from this limit along the edges of the rectangle. We show that in the corner, these fluctuations are gaussian wheras, away from the corner and when the rectangle is a square, the fluctuations are given by the Tracy-Widom distribution. Our method is based on a connection with the Jacobi ensemble.
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.
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.
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.
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）
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.
We prove that the two-dimensional Gaussian Free Field describes the asymptotics of global fluctuations of a multilevel extension of the general beta Jacobi random matrix ensembles. Our approach is based on the connection of the Jacobi ensembles to a degeneration of the Macdonald processes that parallels the degeneration of the Macdonald polynomials to to the Heckman-Opdam hypergeometric functions (of type A). We also discuss the beta goes to infinity limit.
For the random eigenvalues with density corresponding to the Jacobi ensemble $$c \cdot \prod_{i < j} | λ_i - λ_j |^β\prod^n_{i=1} (2 - λ_i)^a (2 + λ_i)^b I_{(-2,2)} (λ_i) $$ $(a, b > -1, β> 0) $ a strong uniform approximation by the roots of the Jacobi polynomials is derived if the parameters $a, b,$ $β$ depend on $n$ and $n \to \infty$. Roughly speaking, the eigenvalues can be uniformly approximated by roots of Jacobi polynomials with parameters $((2a+2)/β-1, (2b+2)/β-1)$, where the error is of order $\{\log n/(a+b) \}^{1/4}$. These results are used to investigate the asymptotic properties of…
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.
We study the global fluctuations for linear statistics of the form $\sum_{i=1}^n f(λ_i)$ as $n \rightarrow \infty$, for $C^1$ functions $f$, and $λ_1, ..., λ_n$ being the eigenvalues of a (general) $β$-Jacobi ensemble, for which tridiagonal models were given by Killip and Nenciu as well as Edelman and Sutton. The fluctuation from the mean ($\sum_{i=1}^n f(λ_i) - \Exp \sum_{i=1}^n f(λ_i)$) is given asymptotically by a Gaussian process. We compute the covariance matrix for the process and show that it is diagonalized by a shifted Chebyshev polynomial basis; in addition, we analyze the deviation …
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…
The paper studies the limiting behavior of spectral measures of random Jacobi matrices of Gaussian, Wishart and MANOVA beta ensembles. We show that the spectral measures converge weakly to a limit distribution which is the semicircle distribution, Marchenko-Pastur distributions or Kesten-Mckey distributions, respectively. The Gaussian fluctuation around the limit is then investigated.
We consider properties of determinants of some random symmetric matrices issued from multivariate statistics: Wishart/Laguerre ensemble (sample covariance matrices), Uniform Gram ensemble (sample correlation matrices) and Jacobi ensemble (MANOVA). If $n$ is the size of the sample, $r\leq n$ the number of variates and $X_{n,r}$ such a matrix, a generalization of the Bartlett-type theorems gives a decomposition of $\det X_{n,r}$ into a product of $r$ independent gamma or beta random variables. For $n$ fixed, we study the evolution as $r$ grows, and then take the limit of large $r$ and $n$ with $…
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 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 express the topological expansion of the Jacobi Unitary Ensemble in terms of triple monotone Hurwitz numbers. This completes the combinatorial interpretation of the topological expansion of the classical unitary invariant matrix ensembles. We also provide effective formulae for generating functions of multipoint correlators of the Jacobi Unitary Ensemble in terms of Wilson polynomials, generalizing the known relations between one point correlators and Wilson polynomials.
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 …
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 $4/β$.
