We introduce a new matrix operation on a pair of matrices, $\text{swirl}(A,X),$ and discuss its implications on the limiting spectral distribution. In a special case, the resultant ensemble converges almost surely to the Rayleigh distribution. In proving this, we provide a novel combinatorial proof that the random matrix ensemble of circulant Hankel matrices converges almost surely to the Rayleigh distribution, using the method of moments.
Limiting distributions of the rightmost eigenvalue of the truncated circular Unitary ensemble across all truncation regimes
School of Mathematical Sciences \& Laboratory of Mathematics；Complex Systems of Ministry of Education, Beijing Normal University, 100875 Beijing, China.
We demonstrate the convergence of the characteristic polynomial of several random matrix ensembles to a limiting universal function, at the microscopic scale. The random matrix ensembles we treat are classical compact groups and the Gaussian Unitary Ensemble. In fact, the result is the by-product of a general limit theorem for the convergence of random entire functions whose zeros present a simple regularity property.
Gyarmati, Mauduit and Sárközy introduced the cross-correlation measure $Φ_k(G)$ of order $k$ to measure the level of pseudorandom properties of families of finite binary sequences. In an earlier paper we estimated the cross-correlation measure of a random family of binary sequences. In this paper, we sharpen these earlier results by showing that for random families, the cross-correlation measure converges strongly, and so has limiting distribution. We also give sharp bounds to the minimum values of the cross-correlation measure, which settles a problem of Gyarmati, Mauduit and Sárközy nearly c…
We study the line ensembles of non-crossing Brownian bridges above a hard wall, each tilted by the area of the region below it with geometrically growing pre-factors. This model, which mimics the level lines of the $(2+1)$D SOS model above a hard wall, was studied in two works from 2019 by Caputo, Ioffe and Wachtel. In those works, the tightness of the law of the top $k$ paths, for any fixed $k$, was established under either zero or free boundary conditions, which in the former setting implied the existence of a limit via a monotonicity argument. Here we address the open problem of a limit und…
Koloğlu, Kopp and Miller compute the limiting spectral distribution of a certain class of real random matrix ensembles, known as $k$-block circulant ensembles, and discover that it is exactly equal to the eigenvalue distribution of an $k \times k$ Gaussian unitary ensemble. We give a simpler proof that under very general conditions which subsume the cases studied by Koloğlu-Kopp-Miller, real-symmetric ensembles with periodic diagonals always have limiting spectral distribution equal to the eigenvalue distribution of a finite Hermitian ensemble with Gaussian entries which is a `complex version'…
We introduce a new family of $N\times N$ random real symmetric matrix ensembles, the $k$-checkerboard matrices, whose limiting spectral measure has two components which can be determined explicitly. All but $k$ eigenvalues are in the bulk, and their behavior, appropriately normalized, converges to the semi-circle as $N\to\infty$; the remaining $k$ are tightly constrained near $N/k$ and their distribution converges to the $k \times k$ hollow GOE ensemble (this is the density arising by modifying the GOE ensemble by forcing all entries on the main diagonal to be zero). Similar results hold for c…
The Limiting Spectral Measure for Ensembles of Symmetric Block Circulant Matrices
给定一个NxN随机矩阵的集合，一个自然的问题是典型矩阵的经验谱测度是否收敛到N --> oo时的极限谱测度。虽然这已经证明了许多薄模式合奏坐在所有实对称矩阵，往往没有很好的封闭形式的表达的限制措施。此外，目前的定理提供了几个图片的合奏之间的过渡。我们考虑元素为i. i. d. r. v.的对称m-块循环矩阵的系综。这些矩阵具有周期为m的环形对角线。我们把m看作是一个“转盘”，我们可以把它从极限特征值密度为高斯的对称循环矩阵的稀疏集合“转”到极限特征值密度为半圆的所有实对称矩阵。当m趋于无穷大时，极限特征值密度f_m显示出视觉上惊人的收敛到半圆，我们证明了这一点。与大多数模式化矩阵系综的研究不同，本文给出了密度的显式封闭形式表达式。证明了f_m是高斯多项式与2 m-2次多项式的乘积，其公式与m × m高斯酉包络（GUE）的公式相同.证据是由时刻。新的功能，使我们能够获得封闭形式的表达式，是转换的中心组合问题的时刻计算到一个等价的代数拓扑计数问题。最后，我们放弃了m个随机变量是不同的假设，推广了m块循环模式。我们证明了极限谱分布的存在，并确定由独立的元素在一个m周期内的模式，不仅取决于每个元素出现的频率，而且元素的排列方式。
We study the limiting occupation density process for a large number of critical and driftless branching random walks. We show that the rescaled occupation densities of $\lfloor sN\rfloor$ branching random walks, viewed as a function-valued, increasing process $\{g_{s}^{N}\}_{s\ge 0}$, converges weakly to a pure jump process in the Skorohod space $\mathbb D([0, +\infty), \mathcal C_{0}(\mathbb R))$, as $N\to\infty$. Moreover, the jumps of the limiting process consist of i.i.d. copies of an Integrated super-Brownian Excursion (ISE) density, rescaled and weighted by the jump sizes in a real-value…
Random Matrix Theory (RMT) has successfully modeled diverse systems, from energy levels of heavy nuclei to zeros of $L$-functions; this correspondence has allowed RMT to successfully predict many number theoretic behaviors. However there are some operations which to date have no RMT analogue. Our motivation is to find an RMT analogue of Rankin-Selberg convolution, which constructs a new $L$-functions from an input pair. We report one such attempt; while it does not appear to model convolution, it does create new ensembles with properties hybridizing those of its constituents. For definiteness …
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.
Random matrix theory successfully models many systems, from the energy levels of heavy nuclei to zeros of $L$-functions. While most ensembles studied have continuous spectral distribution, Burkhardt et al introduced the ensemble of $k$-checkerboard matrices, a variation of Wigner matrices with entries in generalized checkerboard patterns fixed and constant. In this family, $N-k$ of the eigenvalues are of size $O(\sqrt{N})$ and were called bulk while the rest are tightly contrained around a multiple of $N$ and were called blip. We extend their work by allowing the fixed entries to take differen…
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 study the limiting behavior of the $k$-th eigenvalue $x_k$ of unitary invariant ensembles with Freud-type and uniform convex potentials. As both $k$ and $n-k$ tend to infinity, we obtain Gaussian fluctuations for $x_k$ in the bulk and soft edge cases, respectively. Multi-dimensional central limit theorems, as well as moderate deviations, are also proved. This work generalizes earlier results in the GUE and unitary invariant ensembles with monomial potentials of even degree. In particular, we obtain the precise asymptotics of corresponding Christoffel-Darboux kernels as well.
This paper is concerned with the explicit computation of the limiting distribution function of the largest real eigenvalue in the real Ginibre ensemble when each real eigenvalue has been removed independently with constant likelihood. We show that the recently discovered integrable structures in \cite{BB} generalize from the real Ginibre ensemble to its thinned equivalent. Concretely, we express the aforementioned limiting distribution function as a convex combination of two simple Fredholm determinants and connect the same function to the inverse scattering theory of the Zakharov-Shabat syste…
Vinberg cones and the ambient vector spaces are important in modern statistics of sparse models and of graphical models. The aim of this paper is to study eigenvalue distributions of Gaussian, Wigner and covariance matrices related to growing Vinberg matrices, corresponding to growing daisy graphs. For Gaussian or Wigner ensembles, we give an explicit formula for the limiting distribution. For Wishart ensembles defined naturally on Vinberg cones, their limiting Stieltjes transforms, support and atom at 0 are described explicitly in terms of the Lambert-Tsallis functions, which are defined by u…
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…
Consider the class of Ensemble Square Root filtering algorithms for the numerical approximation of the posterior distribution of nonlinear Markovian signals partially observed with linear observations corrupted with independent measurement noise. We analyze the asymptotic behavior of these algorithms in the large ensemble limit both in discrete and continuous time. We identify limiting mean-field processes on the level of the ensemble members, prove corresponding propagation of chaos results and derive associated convergence rates in terms of the ensemble size. In continuous time we also ident…
We consider the symplectic induced Ginibre process, which is a Pfaffian point process on the plane. Let $N$ be the number of points. We focus on the almost-circular regime where most of the points lie in a thin annulus $\mathcal{S}_{N}$ of width $O(\frac{1}{N})$ as $N \to \infty$. Our main results are the scaling limits of all correlation functions near the real axis, and also away from the real axis. Near the real axis, the limiting correlation functions are Pfaffians with a new correlation kernel, which interpolates the limiting kernels in the bulk of the symplectic Ginibre ensemble and of t…
Let $(σ_N^{(i)})_{i \in I}$ be a family of symmetric permutations of the entries of a Wigner matrix $\mathbf{W}_N$. We characterize the limiting traffic distribution of the corresponding family of dependent Wigner matrices $(\mathbf{W}_N^{σ_N^{(i)}})_{i \in I}$ in terms of the geometry of the permutations. We also consider the analogous problem for the limiting joint distribution of $(\mathbf{W}_N^{σ_N^{(i)}})_{i \in I}$. In particular, we obtain a description in terms of semicircular families with general covariance structures. As a special case, we derive necessary and sufficient conditions …
Assume a finite set of complex random variables form a determinantal point process, we obtain a theorem on the limit of the empirical distribution of these random variables. The result is applied to %We study the limits of the empirical distributions of the eigenvalues of two types of $n$ by $n$ random matrices as $n$ goes to infinity. The first one is the product of $m$ i.i.d. (complex) Ginibre ensembles, and the second one is the product of truncations of $m$ independent Haar unitary matrices with sizes $n_j\times n_j$ for $1\leq j \leq m$. Assuming $m$ depends on $n$, by using the special s…
For general $β\geq 1$, we consider Dyson Brownian motion at equilibrium and prove convergence of the extremal particles to an ensemble of continuous sample paths in the limit $N \to \infty$. For each fixed time, this ensemble is distributed as the Airy$_β$ random point field. We prove that the increments of the limiting process are locally Brownian. When $β>1$ we prove that after subtracting a Brownian motion, the sample paths are almost surely locally $r$-H{ö}lder for any $r<1-(1+β)^{-1}$. Furthermore for all $β\geq 1$ we show that the limiting process solves an SDE in a weak sense. When $β=2…
We consider various asymptotic scaling limits $N\to\infty$ for the $2N$ complex eigenvalues of non-Hermitian random matrices in the symmetry class of the symplectic Ginibre ensemble. These are known to be integrable, forming Pfaffian point processes, and we obtain limiting expressions for the corresponding kernel for different potentials. The first part is devoted to the symplectic Ginibre ensemble with the Gaussian potential. We obtain the asymptotic at the edge of the spectrum in the vicinity of the real line. The unifying form of the kernel allows us to make contact with the bulk scaling al…
The classical infinite divisibility of distributions related to eigenvalues of some random matrix ensembles is investigated. It is proved that the $β$-Tracy-Widom distribution, which is the limiting distribution of the largest eigenvalue of a $β$-Hermite ensemble, is not infinitely divisible. Furthermore, for each fixed $N \ge 2$ it is proved that the largest eigenvalue of a GOE/GUE random matrix is not infinitely divisible.
Lattice spin models in statistical physics are used to understand magnetism. Their Hamiltonians are a discrete form of a version of a Dirichlet energy, signifying a relationship to the Harmonic map heat flow equation. The Gibbs distribution, defined with this Hamiltonian, is used in the Metropolis-Hastings (M-H) algorithm to generate dynamics tending towards an equilibrium state. In the limiting situation when the inverse temperature is large, we establish the relationship between the discrete M-H dynamics and the continuous Harmonic map heat flow associated with the Hamiltonian. We show the c…
We characterize the limiting fluctuations of traces of several independent Wigner matrices and deterministic matrices under mild conditions. A CLT holds but in general the families are not asymptotically free of second order and the limiting covariance depends on more information on the deterministic matrices than their limiting *-distribution.
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）
We consider non-Gaussian extensions of the elliptic Ginibre ensemble of complex non-Hermitian random matrices by fixing the trace $\operatorname{Tr}(XX^*)$ of the matrix $X$ with a hard or soft constraint. These ensembles have correlated matrix entries and non-determinantal joint densities of the complex eigenvalues. We study global and local bulk statistics in these ensembles, in particular in the limit of weak non-Hermiticity introduced by Fyodorov, Khoruzhenko and Sommers. Here, the support of the limiting measure collapses to the real line. This limit was motivated by physics applications …
The real Ginibre ensemble consists of $n\times n$ real matrices ${\bf X}$ whose entries are i.i.d. standard normal random variables. In sharp contrast to the complex and quaternion Ginibre ensemble, real eigenvalues in the real Ginibre ensemble attain positive likelihood. In turn, the spectral radius $R_n=\max_{1\leq j\leq n}|z_j({\bf X})|$ of the eigenvalues $z_j({\bf X})\in\mathbb{C}$ of a real Ginibre matrix ${\bf X}$ follows a different limiting law (as $n\rightarrow\infty$) for $z_j({\bf X})\in\mathbb{R}$ than for $z_j({\bf X})\in\mathbb{C}\setminus\mathbb{R}$. Building on previous work b…
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 non-trivial 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 asymptot…
A famous result going back to Eric Kostlan states that the moduli of the eigenvalues of random normal matrices with radial potential are independent yet non identically distributed. This phenomenon is at the heart of the asymptotic analysis of the edge, and leads in particular to the Gumbel fluctuation of the spectral radius when the potential is quadratic. In the present work, we show that a wide variety of laws of fluctuation are possible, beyond the already known cases, including for instance Gumbel and exponential laws at unusual speeds. We study the convergence in law of the spectral radi…
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.
