arXiv subjects

Shirshendu Ganguly

Publications and source records attributed to Shirshendu Ganguly.

At least 73 records · Page 4Linked to original sources

arXiv2018

Entropic CLT and phase transition in high-dimensional Wishart matrices

We consider high dimensional Wishart matrices $\mathbb{X} \mathbb{X}^{\top}$ where the entries of $\mathbb{X} \in {\mathbb{R}^{n \times d}}$ are i.i.d. from a log-concave distribution. We prove an information theoretic phase transition: such matrices are close in total variation distance to the corresponding Gaussian ensemble if and only if $d$ is much larger than $n^3$. Our proof is entropy-based, making use of the chain rule for relative entropy along with the recursive structure in the definition of the Wishart ensemble. The proof crucially relies on the well known relation between Fisher information and entropy, a variational representation for Fisher information, concentration bounds for the spectral norm of a random matrix, and certain small ball probability estimates for log-concave measures.

math.PR↗

arXiv2018

Time Correlation Exponents in Last Passage Percolation

For directed last passage percolation on $\mathbb{Z}^2$ with exponential passage times on the vertices, let $T_{n}$ denote the last passage time from $(0,0)$ to $(n,n)$. We consider asymptotic two point correlation functions of the sequence $T_{n}$. In particular we consider ${\rm Corr}(T_{n}, T_{r})$ for $r\le n$ where $r,n\to \infty$ with $r\ll n$ or $n-r \ll n$. We show that in the former case ${\rm Corr}(T_{n}, T_{r})=Θ((\frac{r}{n})^{1/3})$ whereas in the latter case $1-{\rm Corr}(T_{n}, T_{r})=Θ((\frac{n-r}{n})^{2/3})$. The argument revolves around finer understanding of polymer geometry and is expected to go through for a larger class of integrable models of last passage percolation. As by-products of the proof, we also get a couple of other results of independent interest: Quantitative estimates for locally Brownian nature of pre-limits of Airy$_2$ process coming from exponential LPP, and precise variance estimates for lengths of polymers constrained to be inside thin rectangles at the transversal fluctuation scale.

math.PR↗

arXiv2018

Activated Random Walk on a cycle

We consider Activated Random Walk (ARW), a particle system with mass conservation, on the cycle $\mathbb{Z}/n\mathbb{Z}$. One starts with a mass density $μ>0$ of initially active particles, each of which performs a simple symmetric random walk at rate one and falls asleep at rate $λ>0.$ Sleepy particles become active on coming in contact with other active particles. There have been several recent results concerning fixation/non-fixation of the ARW dynamics on infinite systems depending on the parameters $μ$ and $λ$. On the finite graph $\mathbb{Z}/n\mathbb{Z}$, unless there are more than $n$ particles, the process fixates (reaches an absorbing state) almost surely in finite time. We show that the number of steps the process takes to fixate is linear in $n$ (up to poly-logarithmic terms), when the density is sufficiently low compared to the sleep rate, and exponential in $n$ when the sleep rate is sufficiently small compared to the density, reflecting the fixation/non-fixation phase transition in the corresponding infinite system as established by Rolla, Sidoravicius (2012).

math.PR↗

arXiv2018

Formation of large-scale random structure by competitive erosion

We study the following one-dimensional model of annihilating particles. Beginning with all sites of $\mathbb{Z}$ uncolored, a blue particle performs simple random walk from $0$ until it reaches a nonzero red or uncolored site, and turns that site blue; then, a red particle performs simple random walk from $0$ until it reaches a nonzero blue or uncolored site, and turns that site red. We prove that after $n$ blue and $n$ red particles alternately perform such walks, the total number of colored sites is of order $n^{1/4}$. The resulting random color configuration, after rescaling by $n^{1/4}$ and taking $n\to \infty$, has an explicit description in terms of alternating extrema of Brownian motion (the global maximum on a certain interval, the global minimum attained after that maximum, etc.).

math.PR↗

arXiv2018

Upper tails for arithmetic progressions in a random set

Let $X_k$ denote the number of $k$-term arithmetic progressions in a random subset of $\mathbb{Z}/N\mathbb{Z}$ or $\{1, \dots, N\}$ where every element is included independently with probability $p$. We determine the asymptotics of $\log \mathbb{P}(X_k \ge (1+δ) \mathbb{E} X_k)$ (also known as the large deviation rate) where $p \to 0$ with $p \ge N^{-c_k}$ for some constant $c_k > 0$, which answers a question of Chatterjee and Dembo. The proofs rely on the recent nonlinear large deviation principle of Eldan, which improved on earlier results of Chatterjee and Dembo. Our results complement those of Warnke, who used completely different methods to estimate, for the full range of $p$, the large deviation rate up to a constant factor.

math.PR↗

arXiv2017

Non-fixation for conservative stochastic dynamics on the line

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 $\mathbb{Z}$ with mass conservation. One starts with a mass density $μ>0$ of initially active particles, each of which performs a symmetric random walk at rate one and falls 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 settles a long standing open question.

math.PR↗

arXiv2017

Upper Tail Large Deviations in First Passage Percolation

For first passage percolation on $\mathbb{Z}^2$ with i.i.d. bounded edge weights, we consider the upper tail large deviation event; i.e., the rare situation where the first passage time between two points at distance $n$, is macroscopically larger than typical. It was shown by Kesten (1986) that the probability of this event decays as $\exp (-Θ(n^2))$. However the question of existence of the rate function i.e., whether the log-probability normalized by $n^2$ tends to a limit, had remained open. We show that under some additional mild regularity assumption on the passage time distribution, the rate function for upper tail large deviation indeed exists. Our proof can be generalized to work in higher dimensions and for the corresponding problem in last passage percolation as well. The key intuition behind the proof is that a limiting metric structure which is atypical causes the upper tail large deviation event. The formal argument then relies on an approximate version of the above which allows us to dilate the large deviation environment to compare the upper tail probabilities for various values of $n.$

math.PR↗

arXiv2017

Delocalization of Polymers in Lower Tail Large Deviation

Directed last passage percolation models on the plane, where one studies the weight as well as the geometry of optimizing paths (called polymers) in a field of i.i.d. weights, are paradigm examples of models in the KPZ universality class. In this article, we consider the large deviation regime, i.e., when the polymer has a 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, including the Exponential (Johansson, '00) and Poissonian (Seppäläinen, '98 and Deuschel, Zeitouni, '99) cases. How the geometry of the optimizing paths change under such a large deviation event was considered in (Deuschel, Zeitouni, '99), where it was shown that the paths (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 the corresponding question in the lower tail was left open. We establish a contrasting behavior in the lower tail large deviation regime, showing that conditioned on the latter, in both the models, the optimizing paths are not concentrated around any deterministic curve. Our argument does not use any ingredient from integrable probability, and hence can be extended to other planar last passage percolation models under fairly mild conditions; and also to other non-integrable settings such as high dimensions.

math.PR↗

arXiv2017

Upper tails and independence polynomials in random graphs

The upper tail problem in the Erdős--Rényi random graph $G\sim\mathcal{G}_{n,p}$ 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 showed that in the sparse regime of $p\to 0$ as $n\to\infty$ with $p \geq n^{-α}$ for an explicit $α=α_H>0$, 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 $c_H(δ)$ such that, for $p$ as above and any fixed $δ>0$, the upper tail probability is $\exp[-(c_H(δ)+o(1))n^2 p^Δ\log(1/p)]$, where $Δ$ is the maximum degree of $H$. As it turns out, the leading order constant in the large deviation rate function, $c_H(δ)$, is governed by the independence polynomial of $H$, defined as $P_H(x)=\sum i_H(k) x^k $ where $i_H(k)$ is the number of independent sets of size $k$ in $H$. For instance, if $H$ is a regular graph on $m$ vertices, then $c_H(δ)$ is the minimum between $\frac12 δ^{2/m}$ and the unique positive solution of $P_H(x) = 1+δ$.

math.CO↗

arXiv2017

$SO(N)$ Lattice Gauge Theory, planar and beyond

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 planar setting, which provides certain insights about structures of higher dimensional trajectories as well. Exploiting this, we construct an example showing that in any dimension, the Wilson loop area law lower bound does not hold in full generality.

math.PR↗

arXiv2017

The Competition of Roughness and Curvature in Area-Constrained Polymer Models

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. The three effects arise, for example, in droplets in planar Ising models (Alexander, '01, Hammond, '11,'12). In this article, we offer a new perspective on this phenomenon. We consider directed last passage percolation model in the plane, a paradigmatic example in the KPZ universality class, and constrain the maximizing path under the additional requirement of enclosing an atypically large area. The interface suffers a constraint of parabolic curvature as before, but now its local structure is the KPZ fixed point polymer's rather than Brownian. The local interface fluctuation exponent is thus two-thirds rather than one-half. We prove that the facet lengths of the constrained path's convex hull are governed by an exponent of $3/4$, and inward deviation by an exponent of $1/2$. That is, the exponent triple is now $(2/3,1/2,3/4)$ in place of $(1/2,1/3,2/3)$. 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).

math.PR↗

arXiv2016

Upper triangular matrix walk: Cutoff for finitely many columns

We consider random walk on the group of uni-upper triangular matrices with entries in $\mathbb{F}_2$ 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. 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.

math.PR↗

arXiv2016

Diffusive estimates for random walks on stationary random graphs of polynomial growth

Let $(G,ρ)$ be a stationary random graph, and use $B^G_ρ(r)$ to denote the ball of radius $r$ about $ρ$ in $G$. Suppose that $(G,ρ)$ has annealed polynomial growth, in the sense that $\mathbb{E}[|B^G_ρ(r)|] \leq O(r^k)$ for some $k > 0$ and every $r \geq 1$. Then there is an infinite sequence of times $\{t_n\}$ at which the random walk $\{X_t\}$ on $(G,ρ)$ is at most diffusive: Almost surely (over the choice of $(G,ρ)$), there is a number $C > 0$ such that \[ \mathbb{E} \left[\mathrm{dist}_G(X_0, X_{t_n})^2 \mid X_0 = ρ, (G,ρ)\right]\leq C t_n\qquad \forall n \geq 1\,. \] This result is new even in the case when $G$ is a stationary random subgraph of $\mathbb{Z}^d$. 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.

math.PR↗

arXiv2016

Consistent nonparametric estimation for heavy-tailed sparse graphs

We study graphons as a non-parametric generalization of stochastic block models, and show how to obtain compactly represented estimators for sparse networks in this framework. Our algorithms and analysis go beyond previous work in several ways. First, we relax the usual boundedness assumption for the generating graphon and instead treat arbitrary integrable graphons, so that we can handle networks with long tails in their degree distributions. Second, again motivated by real-world applications, we relax the usual assumption that the graphon is defined on the unit interval, to allow latent position graphs where the latent positions live in a more general space, and we characterize identifiability for these graphons and their underlying position spaces. We analyze three algorithms. The first is a least squares algorithm, which gives an approximation we prove to be consistent for all square-integrable graphons, with errors expressed in terms of the best possible stochastic block model approximation to the generating graphon. Next, we analyze a generalization based on the cut norm, which works for any integrable graphon (not necessarily square-integrable). Finally, we show that clustering based on degrees works whenever the underlying degree distribution is atomless. Unlike the previous two algorithms, this third one runs in polynomial time.

math.ST↗

arXiv2016

Sequence assembly from corrupted shotgun reads

The prevalent technique for DNA sequencing consists of two main steps: shotgun sequencing, where many randomly located fragments, called reads, are extracted from the overall sequence, followed by an assembly algorithm that aims to reconstruct the original sequence. There are many different technologies that generate the reads: widely-used second-generation methods create short reads with low error rates, while emerging third-generation methods create long reads with high error rates. Both error rates and error profiles differ among methods, so reconstruction algorithms are often tailored to specific shotgun sequencing technologies. As these methods change over time, a fundamental question is whether there exist reconstruction algorithms which are robust, i.e., which perform well under a wide range of error distributions. Here we study this question of sequence assembly from corrupted reads. We make no assumption on the types of errors in the reads, but only assume a bound on their magnitude. More precisely, for each read we assume that instead of receiving the true read with no errors, we receive a corrupted read which has edit distance at most $ε$ times the length of the read from the true read. We show that if the reads are long enough and there are sufficiently many of them, then approximate reconstruction is possible: we construct a simple algorithm such that for almost all original sequences the output of the algorithm is a sequence whose edit distance from the original one is at most $O(ε)$ times the length of the original sequence.

q-bio.GN↗

arXiv2015

Convergence of discrete Green functions with Neumann boundary conditions

In this note we prove convergence of Green functions with Neumann boundary conditions for the random walk to their continuous counterparts. Also a few Beurling type hitting estimates are obtained for the random walk on discretizations of smooth domains. These have been used recently in the study of a two dimensional competing aggregation system known as $Competitive\, Erosion$. Some of the statements appearing in this note are classical for ${\mathbb{Z}}^2$. However additional arguments are needed for the proofs in the bounded geometry setting.

math.PR↗

arXiv2015

Heat diffusion with frozen boundary

Consider "Frozen Random Walk" on $\mathbb{Z}$: $n$ particles start at the origin. At any discrete time, the leftmost and rightmost $\lfloor{\frac{n}{4}}\rfloor$ particles are "frozen" and do not move. The rest of the particles in the "bulk" independently jump to the left and right uniformly. The goal of this note is to understand the limit of this process under scaling of mass and time. To this end we study the following deterministic mass splitting process: start with mass $1$ at the origin. At each step the extreme quarter mass on each side is "frozen". The remaining "free" mass in the center evolves according to the discrete heat equation. We establish diffusive behavior of this mass evolution and identify the scaling limit under the assumption of its existence. It is natural to expect the limit to be a truncated Gaussian. A naive guess for the truncation point might be the $1/4$ quantile points on either side of the origin. We show that this is not the case and it is in fact determined by the evolution of the second moment of the mass distribution.

math.PR↗

arXiv2015

Formation of an interface by competitive erosion

In 2006, the fourth author proposed a graph-theoretic model of interface dynamics called competitive erosion. Each vertex of the graph is occupied by a particle that can be either red or blue. New red and blue particles alternately get emitted from their respective bases and perform random walk. On encountering a particle of the opposite color they kill it and occupy its position. We prove that on the cylinder graph (the product of a path and a cycle) an interface spontaneously forms between red and blue and is maintained in a predictable position with high probability.

math.PR↗