Determinantal processes and stochastic domination

Raghavendra Tripathi

Abstract

In this thesis we explore the stochastic domination in determinantal processes. Lyons (2003) showed that if $K_{1}\leq K_{2}$ are two finite rank projection kernels and $P_{1},P_{2}$ are determinantal measures associated with them, then $P_{2}$ stochastically dominates $P_{1}$ , written $P_{1}\prec P_{2},$ that is for every increasing event $\mathcal{A}$ we have $P_{1}(\mathcal{A})\leq P_{2}(\mathcal{A})$ . We give a simpler proof of Lyons’ result which avoids the machinery of exterior algebra used in the original proof of Lyons and also provides a unified approach of proving the result in discrete as well as continuous case.

R. Basu and S. Ganguly (2019) proved the stochastic domination between the largest eigenvalue of Wishart matrix ensemble $W(n,n)$ and $W(n-1,n+1)$ invoking Lyons’ theorem. It is well known that the largest eigenvalue of Wishart ensemble $W(m,n)$ has the same distribution as the directed last-passage time $G(m,n)$ on $\mathbb{Z}^{2}$ with i.i.d. exponential weights. Thus, Basu and Ganguly obtain the stochastic domination between $G(m,n)$ and $G(m-1,n+1).$

It is also known that the largest eigenvalue of the Meixner ensemble $M(m,n)$ has the same distribution as the directed last passage time $G(m,n)$ on $\mathbb{Z}^{2}$ with i.i.d. geometric weights. We prove another stochastic domination result which combined with the Lyons’ theorem gives the stochastic domination between the largest eigenvalues of Meixner ensemble $M(n,n)$ and $M(n-1,n+1),$ which in turn proves that the directed last passage time (with i.i.d. geometric weights) $G(n,n)$ stochastically dominates $G(n-1,n+1).$

{declaration}

I hereby declare that the thesis entitled ‘Determinantal processes and stochastic domination’ submitted by me for the award of M.Sc. degree of the Indian Institute of Science did not form the subject matter for any other thesis submitted by me for any degree or diploma.

Raghavendra Tripathi

SR No:10-06-00-10-31-16-1-13660

Acknowledgements.

Except the mistakes everything else in this work I owe to many people. I am glad to have an opportunity to express my gratitude to those whose help and support I received during this work. First of all, I would like to thank my adviser Prof. Manjunath Krishnapur for his insightful comments, discussions and array of questions during this project. It was his course in Random matrix theory which made me interested in probability, and since then he has continuously helped me navigate my way through vast territory of probability theory. He has very kindly and patiently entertained all my questions and doubts. I learnt great deal of mathematics during my coursework at IISc, and it served to fill many gaps which I would have hardly been able to do on my own. I thank all my instructors for their wonderful courses and their patience to deal with my doubts while I was embarking on a journey to the beautiful world of mathematics. I am also grateful to my fellow students at IISc for many exciting discussions. I am particularly thankful to Abhay Jindal and Shubham Rastogi for being earnest proof readers and pointing out my mistakes– which I make with very high probability. I thank Poornendu Singh and Mayuresh Londhe for their late night discussions and tea– both of which has been equally essential to me. I also thank Mayuresh Londhe for directing me to many new frontiers of mathematics, and the discussions from which I learnt a lot of mathematics which I would have otherwise not known. I would also like to thank my friends in other departments at IISc for their great company. In particular, I must thank Debashree Behera for helping me with innumerable things and keeping me concentrated on my work. I must also thank Prakriti canteen–which has been a quasi-permanent place for my mathematical discussions with friends. A major part of this work was written sitting in Prakriti. While I learnt rigorous mathematics after coming to IISc, I must thank Dr. Mukund Madhav Mishra at Delhi University for his excellent teaching and inspiring me to pursuing mathematics. Equally important were the courses offered by Prof. S. Bagai, Dr. Umesha Kumar, Dr. Yuthika Gadhyan, Dr. Sulbha Arora and many others. Undeniably I owe a lot to my teachers at school and college, my friends and my parents for shaping me as a human being. I can not express my gratitude towards them in mere words. My parents always stood by my side while I was making an excursion from physics to mathematics via engineering, and encouraged to pursue the mathematics. It would have been impossible to come this far without their support. I thank my uncle who was my first math teacher after high school. I also thank my elder sister who helped me continue my education at a time when I felt that I won’t be able to. Needless to say that I am thankful to my friends who have always kept complaining, (for not returning their calls) but who never quit. I hope that they will be happy to see my thesis–even if they do not understand. I have received support from numerous other people, and it would be impossible to name everyone here. But, I thank everyone who has helped and supported me in any direct or indirect ways.

{romanpages}

Chapter 1 Point processes

This chapter aims to provide the background for the upcoming chapters. The primary object of study in this thesis is a determinantal point process and stochatic domination for a special type of determinantal process. Before we specialize to the main theme of the thesis, we will introduce a general point process. There are different possible approaches to introduce the point processes, some of which are specially suitable for specific kind of point processes. The two common approaches to the theory of point process is $a)$ through random sequence of points, and $b)$ through the theory of random measures. In this chapter we briefly describe the two approaches.

In order to give a complete background for the upcoming chapters we will also describe the notion on stochastic domination and coupling in this chapter.

1.1 Definitions and Examples

Roughly speaking, a point process is a probability measure on the space of locally finite configurations in some locally compact Polish space. Much of the theory of the point process is inspired from physics and inadvertently a lot of terminology has been borrowed from physics. The points in a configuration are also referred to as particles. Before we give a rigorous definition of a point process, let us look into some simple examples to get an intuition.

Example 1.

Let $\mathcal{X}$ be a subset of $\mathbb{N}$ which contains every natural number with probability $p$ independently. $\mathcal{X}$ is a random subset of $\mathbb{N}$ . This is an example of a point process.

The above example is of course too simplistic but it contains the key idea that a point-process is simply a random subset of some set. Another simple example of a point process is given below.

Example 2.

Consider a $3\times 3$ matrix with each entry is independently distributed according to a Bernoulli $p$ distribution. And let $\mathcal{X}$ be the set of eigenvalues of such a matrix. It is clear that $\mathcal{X}$ is a random subset of $\mathbb{C}$ , and is an example of a point process.

Note that there are only $2^{9}$ possible matrices in the above example. Using a computer one can explicitly write down all possible values $\mathcal{X}$ takes, with their exact probabilities. Also note that there is nothing special about $3$ , or about the Bernoulli random variables. One can in general start with any random matrix ensemble and the set of eigenvalues will give a point process on $\mathbb{C}.$ We will talk more about such processes later.

With the above two examples we are now prepared to make a definition for the point process. As we have already remarked a (simple) point process on a set $S$ is a random subset of $S.$ Throughout this chapter, we assume $S$ is a locally compact, complete separable metric space (Polish space) equipped with the Borel $\sigma$ -algebra. We start by identifying a random set with a random (Radon) measure on the Borel $\sigma$ -algebra of $S$ . Note that given a locally finite subset $A$ of $S$ , we can associate a measure $\mu_{A}$ on $S$ defined by $\mu_{A}=\sum\limits_{a\in A}\delta_{a}.$ The locally finite assumption on $A$ guarantees that $\mu_{A}$ is a Radon measure. On the other hand, if we have a Radon measure $\eta$ which only takes non-negative integer values (or possibly infinity), then one can similarly associate it with a locally finite configuration (i.e. a multiset) on $S$ . This allows us to see point process as a ‘random variable’ taking values in the space of Radon measures on $S.$ To make this into a formal definition, we shall always take $S$ to be a locally compact Polish space with a reference Radon measure $\mu.$ Denote by $\mathcal{M}(S)$ , the collection of Radon measures on the Borel $\sigma$ -algebra of $S$ which takes values in $\mathbb{N}\cup\{0,\infty\}.$ Equip the collection $\mathcal{M}(S)$ with the vague topology (the topology which $\mathcal{M}(S)$ inherits as the subspace of $C_{0}(S)^{*}$ ), that is, $\mu_{n}\rightarrow\mu$ in $\mathcal{M}(S)$ if $\int fd\mu_{n}\rightarrow\int fd\mu$ for every $f\in C_{0}(S).$

It is well known that $\mathcal{M}(S)$ is a complete separable metric space. This identification allows us to define a point process as a random variable on $(S,\mu)$ taking value in $\mathcal{M}(S).$

Definition 3 (Point Process).

A point process $\mathcal{X}$ on $(S,\mu)$ is a random finite non-negative integer valued Radon measure on $S.$ It is called a simple point process if $\mathcal{X}(\{s\})\leq 1$ for every $s\in S,$ almost surely.

It is instructive to think of a simple point process as a random discrete subset of $S.$ It should be pointed out that by the definition of the simple point process, $\mathcal{X}(D)$ is the random variable which counts the number of points (or particles) in the set $D$ , for any Borel subset $D\subset S$ . The measurability of $\mathcal{X}$ turns out to be equivalent to the measurability of random variables $\mathcal{X}(D)$ for every Borel subset $D\subset S.$

Let us explore a few more examples to understand these point processes better.

Example 4 (Discrete Poisson process).

Let $S$ be a finite or countable set with a Radon measure $\mu.$ And let $X$ be random multiset of $S$ where the multiplicity of each $x\in\mathcal{X}$ is an independent Poisson with intensity $\mu\{x\}.$ Equivalently $\mathcal{X}$ is random measure defined as $\sum\limits_{x\in S}P_{x}\delta_{x},$ where $P_{x},x\in S,$ are independent random variables and $P_{x}\sim\mbox{Pois}(\mu\{x\}).$

The above example also affords us an example of non-simple point process. We do have a continuous analogue of the above process which we record below with a caution that the existence of a process with the properties described below is not at all immediate. We refer the interested reader to [Jones].

Example 5 (General Poisson process).

Let $S$ be a locally compact Polish space with a Radon measure $\mu.$ Let $\mathcal{X}$ be the process such that for any $A\subset S$ of finite measure, the number of points in $\mathcal{X}(A)$ is distributed by Poisson random variable $P_{A}$ with intensity $\mu(A)\leq\infty.$ And for any collection of disjoint subsets $A_{1},A_{2},\dots,A_{k}$ of finite measure the collection of random variables $\{P_{A_{i}}:1\leq i\leq k\}$ is independent.

We now turn towards the question of describing a point process. Inspired by the general theory of stochastic processes, one would imagine that the natural way to describe a point process would be by describing the probabilities of its cylinder sets i.e. by specifying the $\Pr[\mathcal{X}(B_{i})=k_{i},1\leq i\leq m]$ for all $m\geq 1$ and Borel subsets $B_{i}\subset S$ . Of course, in order to define a point process the assignment of probabilities to the cylinder sets must be consistent meaning that

\sum\limits_{0\leq k_{m+1}\leq\infty}\Pr[\mathcal{X}(B_{i})=k_{i},1\leq i\leq m+1]=\Pr[\mathcal{X}(B_{i})=k_{i},1\leq i\leq m].

This indeed is useful and very much in the spirit of general theory of stochastic processes. But this is not the most preferred or the most amenable way to describe a point process. The distribution of a point process is most often described by its joint intensities/correlation functions. Of course, there are other ways to describe a point process but we will not get into details here. We also caution the reader the joint intensities do not always exist and even when they do, they need not completely determine a point process, but for all our purposes specifying the joint intensities would be enough. For a short but beautiful discussion of joint intensities we suggest the reader to look into Chapter 1 of [GAF], and also the survey paper [Peres], which contains everything necessary for our purposes. For a full treatment of theory of point-process and understanding full nuances, we also refer the reader to [Jones]. Here we content ourselves with the definitions and facts that would be useful to us later. Recall that $(S,\mu)$ is a locally compact Polish space equipped with the Borel $\sigma$ -algebra and $\mu$ is a Radon measure on $S$ .