arXiv subjects

Manjunath Krishnapur

Publications and source records attributed to Manjunath Krishnapur.

28 records · Page 2Linked to original sources

arXiv2011

The Ginibre ensemble and Gaussian analytic functions

We show that as $n$ changes, the characteristic polynomial of the $n\times n$ random matrix with i.i.d. complex Gaussian entries can be described recursively through a process analogous to Pólya's urn scheme. As a result, we get a random analytic function in the limit, which is given by a mixture of Gaussian analytic functions. This gives another reason why the zeros of Gaussian analytic functions and the Ginibre ensemble exhibit similar local repulsion, but different global behavior. Our approach gives new explicit formulas for the limiting analytic function.

math.PR↗

arXiv2011

Lipschitz correspondence between metric measure spaces and random distance matrices

Given a metric space with a Borel probability measure, for each integer $N$ we obtain a probability distribution on $N\times N$ distance matrices by considering the distances between pairs of points in a sample consisting of $N$ points chosen indepenedently from the metric space with respect to the given measure. We show that this gives an asymptotically bi-Lipschitz relation between metric measure spaces and the corresponding distance matrices. This is an effective version of a result of Vershik that metric measure spaces are determined by associated distributions on infinite random matrices.

math.PR↗

arXiv2010

The single ring theorem

We study the empirical measure $L_{A_n}$ of the eigenvalues of non-normal square matrices of the form $A_n=U_nD_nV_n$ with $U_n,V_n$ independent Haar distributed on the unitary group and $D_n$ real diagonal. We show that when the empirical measure of the eigenvalues of $D_n$ converges, and $D_n$ satisfies some technical conditions, $L_{A_n}$ converges towards a rotationally invariant measure on the complex plan whose support is a single ring. In particular, we provide a complete proof of Feinberg-Zee single ring theorem \cite{FZ}. We also consider the case where $U_n,V_n$ are independent Haar distributed on the orthogonal group.

math.PR↗

arXiv2009

Derivation of an eigenvalue probability density function relating to the Poincare disk

A result of Zyczkowski and Sommers [J.Phys.A, 33, 2045--2057 (2000)] gives the eigenvalue probability density function for the top N x N sub-block of a Haar distributed matrix from U(N+n). In the case n \ge N, we rederive this result, starting from knowledge of the distribution of the sub-blocks, introducing the Schur decomposition, and integrating over all variables except the eigenvalues. The integration is done by identifying a recursive structure which reduces the dimension. This approach is inspired by an analogous approach which has been recently applied to determine the eigenvalue probability density function for random matrices A^{-1} B, where A and B are random matrices with entries standard complex normals. We relate the eigenvalue distribution of the sub-blocks to a many body quantum state, and to the one-component plasma, on the pseudosphere.

math-ph↗

arXiv2009

Random matrices: Universality of ESDs and the circular law

Given an $n \times n$ complex matrix $A$, let $$μ_{A}(x,y):= \frac{1}{n} |\{1\le i \le n, \Re λ_i \le x, \Im λ_i \le y\}|$$ be the empirical spectral distribution (ESD) of its eigenvalues $λ_i \in \BBC, i=1, ... n$. We consider the limiting distribution (both in probability and in the almost sure convergence sense) of the normalized ESD $μ_{\frac{1}{\sqrt{n}} A_n}$ of a random matrix $A_n = (a_{ij})_{1 \leq i,j \leq n}$ where the random variables $a_{ij} - \E(a_{ij})$ are iid copies of a fixed random variable $x$ with unit variance. We prove a \emph{universality principle} for such ensembles, namely that the limit distribution in question is {\it independent} of the actual choice of $x$. In particular, in order to compute this distribution, one can assume that $x$ is real of complex gaussian. As a related result, we show how laws for this ESD follow from laws for the \emph{singular} value distribution of $\frac{1}{\sqrt{n}} A_n - zI$ for complex $z$. As a corollary we establish the Circular Law conjecture (in both strong and weak forms), that asserts that $μ_{\frac{1}{\sqrt{n}} A_n}$ converges to the uniform measure on the unit disk when the $a_{ij}$ have zero mean.

math.PR↗

arXiv2007

From random matrices to random analytic functions

We consider two families of random matrix-valued analytic functions: (1) G_1-zG_2 and (2) G_0 + zG_1 +z^2G_2+ ..., where G_i are n x n independent random matrices with independent standard complex Gaussian entries. The set of z where these matrix-valued analytic functions become singular, are shown to be determinantal point processes on the sphere and the hyperbolic plane, respectively. The kernels of these determinantal processes are reproducing kernels of certain natural Hilbert spaces of analytic functions on the corresponding surfaces. This gives a unified framework in which to view a result of Peres and Virag (n=1 in the second setting) and a well known theorem of Ginibre on Gaussian random matrices (which may be viewed as an analogue of our results in the whole plane).

math.PR↗

arXiv2006

Zeros of Random Analytic Functions

The dominant theme of this thesis is that random matrix valued analytic functions, generalizing both random matrices and random analytic functions, for many purposes can (and perhaps should) be effectively studied in that level of generality. We study zeros of random analytic functions in one complex variable. It is known that there is a one parameter family of Gaussian analytic functions with zero sets that are stationary in each of the three symmetric spaces, namely the plane, the sphere and the unit disk, under the corresponding group of isometries. We show a way to generate non Gaussian random analytic functions whose zero sets are also stationary in the same domains. There are particular cases where the exact distribution of the zero set turns out to belong to an important class of point processes known as determinantal point processes. Apart from questions regarding the exact distribution of zero sets, we also study certain asymptotic properties. We show asymptotic normality for smooth statistics applied to zeros of these random analytic functions. Lastly, we present some results on certain large deviation problems for the zeros of the planar and hyperbolic Gaussian analytic functions.

math.PR↗

arXiv2006

Determinantal Processes and Independence

We give a probabilistic introduction to determinantal and permanental point processes. Determinantal processes arise in physics (fermions, eigenvalues of random matrices) and in combinatorics (nonintersecting paths, random spanning trees). They have the striking property that the number of points in a region $D$ is a sum of independent Bernoulli random variables, with parameters which are eigenvalues of the relevant operator on $L^2(D)$. Moreover, any determinantal process can be represented as a mixture of determinantal projection processes. We give a simple explanation for these known facts, and establish analogous representations for permanental processes, with geometric variables replacing the Bernoulli variables. These representations lead to simple proofs of existence criteria and central limit theorems, and unify known results on the distribution of absolute values in certain processes with radially symmetric distributions.

math.PR↗

arXiv2005

Overcrowding estimates for zeroes of Planar and Hyperbolic Gaussian analytic functions

We consider the point process of zeroes of certain Gaussian analytic functions and find the asymptotics for the probability that there are more than m points of the process in a fixed disk of radius r, as m-->infinity. For the Planar Gaussian analytic function, sum_n a_n z^n/sqrt(n!), we show that this probability is asymptotic to exp(-0.5 m^2 log(m)). For the Hyperbolic Gaussian analytic functions, sum_n sqrt({-rho choose n}) a_n z^n, rho>0, we show that this probability decays like exp(-cm^2). In the planar case, we also consider the problem posed by Mikhail Sodin on moderate and very large deviations in a disk of radius r as r --> infinity. We partly solve the problem by showing that there is a qualitative change in the asymptotics of the probability as we move from the large deviation regime to the moderate.

math.PR↗

arXiv2004

Recurrent graphs where two independent random walks collide finitely often

We present a class of graphs where simple random walk is recurrent, yet two independent walkers meet only finitely many times almost surely. In particular, the comb lattice, obtained from Z^2 by removing all horizontal edges off the X-axis, has this property. We also conjecture that the same property holds for some other graphs, including the incipient infinite cluster for critical percolation in Z^2.

math.PR↗