In honour of Rajeeva L. Karandikar
A conference on Probability & Stochastic Processes
29 - 31 March 2022

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Lecture Abstracts

Amarjit Budhiraja, University of North Carolina
at Chapel Hill

Domains of attraction of invariant distributions of the infinite Atlas model

The infinite Atlas model describes a countable system of competing Brownian particles where the lowest particle gets a unit upward drift and the rest evolve as standard Brownian motions. The stochastic process of gaps between the particles in the infinite Atlas model does not have a unique stationary distribution and in fact for every \(a \ge 0\), \(\pi_a := \bigotimes_{i=1}^{\infty} \operatorname{Exp}(2 + ia)\) is a stationary measure for the gap process. We say that an initial distribution of gaps is in the weak domain of attraction of the stationary measure \(\pi_a\) if the time averaged laws of the stochastic process of the gaps, when initialized using that distribution, converge to \(\pi_a\) weakly in the large time limit. We provide general sufficient conditions on the initial gap distribution of the Atlas particles for it to lie in the weak domain of attraction of \(\pi_a\) for each \(a\ge 0\).

This is joint work with Sayan Banerjee.

Arup Bose, Indian Statistical Institute, Kolkata

Large Dimensional Random Matrices and High Dimensional Statistics

We shall discuss some results on Large Dimensional Random Matrices that have been obtained in recent years, especially in the context of High Dimensional Statistics.

Kumarjit Saha, Ashoka University

Convergence to the Brownian web for a perturbed Howard model

Understanding the structure of random directed forests constructed on random sets of points and studying their scaling limits has been extensively studied in the literature. For most models, the corresponding random sets of points satisfy independence assumption over disjoint regions. A natural question is what happens for point processes if such an assumption does not hold.

In a joint work with Subhroshekhar Ghosh, we explored such questions where lattice points are perturbed over compact regions. In this talk I will describe a recent work with Rahul Roy and Anish Sarkar where we are dealing with perturbed lattice points and perturbations are no longer restricted to compact regions. The dependency of the generated point process is quite challenging to deal with and requires non-trivial modification of existing techniques.

M. Vidyasagar, Indian Institute of Technology
Hyderabad

Convergence of stochastic approximation algorithms via martingale and converse Lyapunov methods

Ever since its introduction in 1951 by Robbins and Monro, the stochastic approximation (SA) algorithm has been a workhorse for establishing the convergence of stochastic algorithms in various contexts. In recent times, these ideas are being widely used in Reinforcement Learning (RL), one of the most active areas in Machine Learning (ML) and AI. The assumptions of "standard" SA are not always satisfied in these RL applications; hence some modifications in the theory are required. The usual convergence proofs of SA are based on the so-called ODE method, whereby it is shown that the sample paths of the algorithm approach the (deterministic) solution trajectories of an associated ODE. In this talk I will propose an entirely different approach based on martingale theory, building on ideas first proposed by Gladyshev in 1965. I will first state and prove two simple technical lemmas that capture the essence of the SA approach. Then I will state (but not prove in the talk) a new "converse Lyapunov theorem" for the stability of nonlinear ODEs. Taken together, these two results allow one to find a very simple proof of the convergence of SA in a variety of settings. Towards the end of the talk, I will discuss extensions of standard SA to problems in RL.

Manjunath Krishnapur, Indian Institute of Science
Bangalore

Random analytic functions from orthogonal polynomials on the unit circle

Peres and Virag (2005) proved that the power series with i.i.d. complex Gaussian coefficients has a zero set that is a conformally invariant determinantal point process on the plane, determined by the Bergman kernel. Generalizing this, it was proved that the singular points of the matrix power series with i.i.d. complex Gaussian matrices (Ginibre matrices) also forms a determinantal point process with more general Bergman-like kernels on the disk. Earlier proofs of this result relied on a result of Zyczkowski and Sommers on truncated Haar unitary matrices. In this talk we describe a new and more direct proof of the theorem using orthogonal polynomials on the unit circle.

Mathew Joseph, Indian Statistical Institute, Bangalore

Small-ball probabilities for the stochastic heat equation

We consider the stochastic heat equation (SHE) on an interval with periodic boundary conditions. We discuss the small-ball probability for the SHE, the probability that the SHE is unusually small (lies in a small ball) up to a fixed time. We do this for the sup norm and the Holder semi norms. Under fairly general conditions we are able to provide near-optimal bounds on these probabilities. This talk is based on joint works with Siva Athreya and Carl Mueller, and with Mohammud Foondun and Kunwoo Kim.

Probal Chaudhuri, Indian Statistical Institute, Kolkata

Mahalanobis' Distance : A Brief History and Some Observations

Mahalanobis distance plays a critical role in statistical discriminant analysis. This talk will present a brief review of the history of Mahalanobis distance and some not so well known yet interesting facts about it.

Riddhipratim Basu, International Centre for Theoretical
Sciences, Bengaluru

First passage percolation on hyperbolic groups

First passage percolation is a canonical model of random metric where each edge of a graph is assigned an independent and identically distributed non-negative random length and one studies the properties of the resulting metric and the associated geodesics. This model is extensively studied on the Euclidean lattice, but other geometries have been considered as well. We consider first passage percolation on Cayley graphs of Gromov hyperbolic groups and answer several natural questions including the first and second order behaviour of the passage time along word geodesics, and the direction and coalsences of the random geodesics.

This is a joint work with Mahan Mj.

Shirshendu Ganguly, University of California, Berkeley

Fractal geometry in models of random growth

In last passage percolation models predicted to lie in the Kardar-Parisi-Zhang (KPZ) universality class, geodesics are oriented paths moving through random noise accruing maximum weight. The weights of such geodesics as their endpoints are varied give rise to an intricate random energy field expected to converge to a rich universal object known as the Directed Landscape constructed by Dauvergne, Ortmann and Virag.

The Directed Landscape satisfies various scale invariance properties making it a source of fascinating random fractal behavior which has been investigated intensely since its construction. In this talk, we will report recent progress in this direction. In particular, we will introduce the notion of a ``local time" for geodesics and present a sketch of its construction, and further use that to compute new fractal exponents governing aspects of the ``temporal geometry" of the Directed Landscape.

The talk will be based on a recent work with L. Zhang.

Sourav Chatterjee, Standford University

A new coefficient of correlation

Is it possible to define a coefficient of correlation which is (a) as simple as the classical coefficients like Pearson's correlation or Spearman's correlation, and yet (b) consistently estimates some simple and interpretable measure of the degree of dependence between the variables, which is 0 if and only if the variables are independent and 1 if and only if one is a measurable function of the other, and (c) has a simple asymptotic theory under the hypothesis of independence, like the classical coefficients? I will talk about a recent work that answers this question in the affirmative, by producing such a coefficient. No assumptions are needed on the distributions of the variables.