Shirshendu GangulyI am currently a Miller Fellow in Statistics and Mathematics at UC Berkeley. I got my PhD in Mathematics from University of Washington in Spring 2016 where I was advised by Ioana Dumitriu and Christopher Hoffman . Prior to this, I obtained bachelors and masters in Statistics from Indian Statistical Institute, Kolkata. 
ResearchI am broadly interested in probability theory and its applications. Recently I have been working on problems in markov chains related to spin systems, random walk on graphs, Gibbs measure on random matrices, geometry of geodesics in percolation, and large deviation theory in sparse combinatorial settings, etc. I am also interested in applications of information theory to understand high dimensional phenomena.
A complete list of publications and preprints is given here. Here is my CV. 
Research Internships 
Some Recent Works 
Mixing time of random walk on matrices. We consider random walk on the group of uniupper triangular matrices with entries in F_2 which forms an important example of a nilpotent group. Peres and Sly (2013) proved tight bounds on the mixing time of this walk up to constants. It is well known that the single column projection of this chain is the one dimensional East process. In this article, we complement the PeresSly result by proving a cutoff result for the mixing of finitely many columns in the upper triangular matrix walk at the same location as the East process of the same dimension. Moreover, we also show that the spectral gaps of the matrix walk and the East process are equal. The proof of the cutoff result is based on a recursive argument which uses a local version of a dual process appearing in Peres and Sly (2013), various combinatorial consequences of mixing and concentration results for the movement of the front in the one dimensional East process. 
Escape rates of Random walk on stationary graphs. We show that for any stationary random graph (G,p) of annealed polynomial growth there is an infinite sequence of times {t_n} at which the random walk {X_t} on (G,p) is at most diffusive. This result is new even in the case when G is a stationary random subgraph of Z^d. Combined with the work of Benjamini, DuminilCopin, Kozma, and Yadin (2015), it implies that G almost surely does not admit a nonconstant harmonic function of sublinear growth. To complement this, we argue that passing to a subsequence of times {t_n} is necessary, as there are stationary random graphs of (almost sure) polynomial growth where the random walk is almost surely superdiffusive at an infinite subset of times. 
Gibbs measure on Random Matrices and Lattice gauge theory. Lattice Gauge theories have been studied in the physics literature as discrete approximations to quantum YangMills theory for a long time. Primary statistics of interest in these models are expectations of the so called “Wilson loop variables”. In this article we continue the program initiated by Chatterjee (2015) to understand Wilson loop expectations in Lattice Gauge theories in a certain limit through gaugestring duality. The objective in this paper is to better understand the underlying combinatorics in the strong coupling regime, by giving a more geometric picture of string trajectories involving correspondence to objects such as decorated trees and noncrossing partitions. Using connections with Free Probability theory, we provide an elaborate description of loop expectations in the the planar setting, which provides certain insights about structures of higher dimensional trajectories as well. Exploiting this, we construct a counter example showing that in any dimension, the Wilson loop area law lower bound does not hold in full generality, and thereby partially answer a question from Chatterjee (2015) in the negative. 
Competitive Erosion Competitive erosion is conformally invariant (with Yuval Peres) Arxiv We study a graphtheoretic model of interface dynamics called Competitive Erosion. Each vertex of the graph is occupied by a particle, which can be either red or blue. New red and blue particles are emitted alternately from their respective bases and perform random walk. On encountering a particle of the opposite color they remove it and occupy its position. Based on conformally invariant nature of reflected brownian motion Propp conjectured in 2003 that at stationarity, with high probability the blue and the red regions are separated by an orthogonal circular arc on the disc and by a suitable hyperbolic geodesic on a general simply connected domain. In the first article we study Competitive Erosion on the cylinder graph which has a relatively simple geometry. In the second article we confirm Propp's prediction on smooth enough planar simply connected domains. 
Large deviation and counting problems in sparse settings. Upper tails for arithmetic progressions in a random set (with with Bhaswar B. Bhattacharya, Xuancheng Shao, Yufei Zhao) Arxiv The upper tail problem in the ErdosRenyi random graph G∼G(n,p) asks to estimate the probability that the number of copies of a graph H in G exceeds its expectation by a factor 1+δ. Chatterjee and Dembo (2014) showed that in the sparse regime of p→0 as n→∞ with p bigger than an inverse power of n, this problem reduces to a natural variational problem on weighted graphs, which was thereafter asymptotically solved by two of the authors in the case where H is a clique. Here we extend the latter work to any fixed graph H and determine a function cH(δ) such that, for p as above and any fixed δ >0, the logarithm of the upper tail probability properly normalized converges to cH(δ). As it turns out, the leading order constant in the large deviation rate function, cH(δ), is governed by the independence polynomial of H. The second article investigates similar questions in the setting of counting arithmetic progressions in random subsets of Z/nZ. For progressions of length greater than three, a new large deviation principle is established using the arithmetic regularity lemma based on the powerful machinery of inverse theorem for Gowers norms. 
Self organized criticality We consider Activated Random Walk (ARW), a model which generalizes the Stochastic Sandpile, one of the canonical examples of self organized criticality. Informally ARW is a particle system on Z, with initial mass density μ > 0 of active particles. Active particles do a symmetric random walk at rate one and fall asleep at rate λ > 0. Sleepy particles become active on coming in contact with other active particles. We investigate the question of fixation/nonfixation of the process and show for small enough λ the critical mass density for fixation is strictly less than one. Moreover, the critical density goes to zero as λ tends to zero. This positively answers two open questions from Dickman, Rolla, Sidoravicius (J. Stat. Phys., 2010) and Rolla, Sidoravicius (Invent. Math., 2012).
The second figure shows the set of occupied sites on the two dimensional lattice.
