Shirshendu Ganguly

Shirshendu Ganguly


I 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.

Me

Research

I am broadly interested in probability theory and its applications. Recently I have been working on problems in Disordered metric geometries with focus on geometry of geodesics in percolation models, Scaling limits and Phase transitions in statistical mechanics, Large deviations and counting problems in sparse non-linear settings, Mixing time of Markov Chains, Random walk on graphs and Random Matrix theory.

A complete list of publications and preprints is given here. Here is my CV.

Research Internships

  • Microsoft Research, Redmond: Summer, 2013. Mentored by Eyal Lubetzky and Yuval Peres.
  • Microsoft Research, NE: Summer, 2014. Mentored by Christian Borgs.
  • Microsoft Research, Redmond: Summer, 2015. Mentored by Sebastien Bubeck.
  • Some Recent Works

    Fluctuation exponents, delocalization and large deviation of polymers

    The Competition of Roughness and Curvature in Area-Constrained Polymer Models (with Riddhipratim Basu, Alan Hammond) Arxiv
    Delocalization of polymers in lower tail large deviation. (with Riddhipratim Basu, Allan Sly) Arxiv
    Upper tail large deviations in First Passage Percolation. (with Riddhipratim Basu, Allan Sly) Draft available upon request.


    The competition between local Brownian roughness and global parabolic curvature experienced in many random interface models reflects an important aspect of the KPZ universality class. It may be summarised by an exponent triple (1/2,1/3,2/3) representing local interface fluctuation, local roughness (or inward deviation) and convex hull facet length. In the first article, we offer a new perspective on this phenomenon. We consider directed last passage percolation model in the plane, and constrain the maximizing path under the additional requirement of enclosing an atypically large area. We prove that the exponent triple is now (2/3,1/2,3/4). This phenomenon appears to be shared among various isoperimetrically extremal circuits in local randomness. Indeed, we formulate a conjecture to this effect concerning such circuits in supercritical percolation, whose Wulff-like first-order behaviour was recently established (Biskup, Louidor, Procaccia and Rosenthal, '12).

    In the second article, we consider the large deviation regime, i.e., when the geodesic has much smaller (lower tail) or larger (upper tail) weight than typical. Precise asymptotics of large deviation probabilities have been obtained in a handful of the so-called exactly solvable scenarios. How the geometry of the geodesic changes under such a large deviation event was considered by Dueschel and Zeitouni (1998) where it was shown that the geodesic (from (0,0) to (n,n), say) remain concentrated around the straight line joining the end points in the upper tail large deviation regime, but left open the corresponding question in the lower tail. We answer their question by establishing a contrasting behaviour in the lower tail large deviation regime; we show that conditioned on the latter, in both the models, the geodesic is not concentrated around any deterministic curve. Our argument does not use any ingredient from integrable probability.

    In the third article we establish a precise large deviation principle for the upper tail in First passage percolation which had remained open since Kesten's seminal work in 1986 which established the asymptotic decay rate of probabilities.

    The geodesic approximately passes through an uniformly chosen point on the anti-diagonal in large deviation lower tail regime in LPP causing it to be delocalized.

    Escape rates of Random walk on stationary graphs.

    Diffusive estimates for random walks on stationary random graphs of polynomial growth. (with James R. Lee, Yuval Peres) To appear in GAFA (2017) Arxiv

    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, Duminil-Copin, Kozma, and Yadin (2015), it implies that G almost surely does not admit a non-constant 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.
    Example of a tree of `stretched' expanders where at infinite sequence of times random walk is superdiffusive.

    Gibbs measure on Random Matrices and Lattice gauge theory.

    SO(N) Lattice Gauge Theory, Planar and Beyond (with Riddhipratim Basu) To appear in Communications on Pure and Applied Mathematics (2017). Arxiv

    Lattice Gauge theories have been studied in the physics literature as discrete approximations to quantum Yang-Mills 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 gauge-string 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 non-crossing 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.

    Competitive Erosion

    Interface formation by Competitive Erosion (with Lionel Levine, Yuval Peres and James Propp) To appear in PTRF (2016) Arxiv
    Competitive erosion is conformally invariant (with Yuval Peres) Arxiv
    Formation of large scale random structure by competitive erosion. (with Lionel Levine and Sourav Sarkar) Draft available upon request.

    We study a graph-theoretic 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.
    In the third article we study a variant on the line with the same source for both the particles which exhibit a random interface dynamics unlike the former cases.
    Competitive erosion started from a random coloring spontaneously forms an interface.

    Large deviation and counting problems in sparse settings.

    Upper tails and independence polynomials in random graphs (with Bhaswar B. Bhattacharya, Eyal Lubetzky, Yufei Zhao). To appear in Advances in Mathematics (2017). Arxiv
    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 Erdos-Renyi 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 reducing large deviation questions to extremal problems in additive combinatorics.

    Self organized criticality

    Non-fixation for conservative stochastic dynamics on the line (with Riddhipratim Basu, Christopher Hoffman). To appear in Communications in Mathematical Physics (2017). Arxiv
    Activated Random walk on a cycle. (with Riddhipratim Basu, Christopher Hoffman and Jacob Richey). Arxiv

    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/non-fixation 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).

    East Process and random walk on matrices

    Cutoff for the East Process (with Eyal Lubetzky and Fabio Martinelli) Communications in Mathematical Physics (2015). Arxiv
    Upper triangular matrix walk: cutoff for finitely many columns. (with Fabio Martinelli) Arxiv


    The East process is a 1D kinetically constrained interacting particle system, introduced in the physics literature in the early 90's to model liquid-glass transitions. Spectral gap estimates of Aldous and Diaconis in 2002 imply that its mixing time on L sites has order L. We complement that result and show cutoff with an O(√L)-window. Finally, we supplement these results by a study of analogous kinetically constrained models on trees, again establishing cutoff, yet this time with an O(1)-window.
    In the second article we consider random walk on the group of uni-upper 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 Peres-Sly 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.

    Coauthors

    Riddhipratim Basu , Christian Borgs , Bhaswar Bhattacharya, Sébastien Bubeck , Elisabetta Candellero , Jennifer. T. Chayes , Henry Cohn , Ioana Dumitriu, Laura Florescu , Alan Hammond , Christopher Hoffman , James R. Lee , Lionel Levine , Eyal Lubetzky , Fabio Martinelli , Soumik Pal , Yuval Peres , James Propp , Fernando Xuancheng Shao , Joel Spencer , Allan Sly , Yufei Zhao .

    Teaching

    I have been a teaching assistant for undergraduate calculus courses in Fall 2011, Winter, Spring 2012, and Winter 2015 at UW. I have also been a grader for Graduate Complex Analysis in Fall, Winter 2013 and for Graduate Real Analysis in Fall 2015, Winter and Spring 2016.

    Contact

    307, Evans Hall,
    Department of Statistics,
    University of California, Berkeley,
    CA, 94720.
    Email: sganguly@berkeley.edu.