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Antar Bandyopadhyay

Publications and source records attributed to Antar Bandyopadhyay.

25 records · Page 2Linked to original sources
arXiv2006

Random Walk in Dynamic Markovian Random Environment

We consider a model, introduced by Boldrighini, Minlos and Pellegrinotti, of random walks in dynamical random environments on the integer lattice Z^d with d>=1. In this model, the environment changes over time in a Markovian manner, independently across sites, while the walker uses the environment at its current location in order to make the next transition. In contrast with the cluster expansions approach of Boldrighini, Minlos and Pellegrinotti, we follow a probabilistic argument based on regeneration times. We prove an annealed SLLN and invariance principle for any dimension, and provide a quenched invariance principle for dimension d > 7, providing for d>7 an alternative to the analytical approach of Boldrighini, Minlos and Pellegrinotti, with the added benefit that it is valid under weaker assumptions. The quenched results use, in addition to the regeneration times already mentioned, a technique introduced by Bolthausen and Sznitman.

math.PR↗
arXiv2006

Bivariate Uniqueness and Endogeny for Recursive Distributional Equations : Two Examples

This paper has been withdrawn by the author because of finding a flaw in the proof of endogeny for the Frozen Percolation RDE which was one of two examples discussed in this paper. The other example is correct and can be obtained from the work "Bivariate Uniqueness and Endogeny for the Logistic Recursive Distributional Equation" of the author. At the current moment we believe that endogeny is false for the Frozen Percolation RDE. Fresh simulations done independently by the author and David J. Aldous suggest non-endogeny. Although a rigorous proof is yet to be found. It is interesting to note that one can prove that the tail of the associated Recursive Tree Process (RTP) is trivial. The work titled "A Necessary and Sufficient Condition for the Tail Triviality of a Recursive Tree Process" of the author present this result.

math.PR↗
arXiv2006

A Necessary and Sufficient Condition for the Tail-Triviality of a Recursive Tree Process

Given a recursive distributional equation (RDE) and a solution $μ$ of it, we consider the tree indexed invariant process called the recursive tree process (RTP) with marginal $μ$. We introduce a new type of bivariate uniqueness property which is different from the one defined by Aldous and Bandyopadhyay (2005), and we prove that this property is equivalent to tail-triviality for the RTP, thus obtaining a necessary and sufficient condition to determine tail-triviality for a RTP in general. As an application we consider Aldous' (2000) construction of the frozen percolation process on a infinite regular tree and show that the associated RTP has a trivial tail.

math.PR↗
arXiv2006

Bivariate Uniqueness and Endogeny for the Logistic Recursive Distributional Equation

In this article we prove the bivariate uniqueness property for a particular ``max-type'' recursive distributional equation (RDE). Using the general theory developed by Aldous and Bandyopadhyay (2005) we then show that the corresponding recursive tree process (RTP) has no external randomness, more preciously, the RTP is endogenous. The RDE we consider is so called the Logistic RDE, which appears in Aldous' (2001) proof of the $ζ(2)$-limit of the random assignment problem using the local weak convergence method. Thus this work provides a non-trivial application of the general theory developed by Aldous and Bandyopadhyay (2005).

math.PR↗
arXiv2005

Counting without sampling. New algorithms for enumeration problems using statistical physics

We propose a new type of approximate counting algorithms for the problems of enumerating the number of independent sets and proper colorings in low degree graphs with large girth. Our algorithms are not based on a commonly used Markov chain technique, but rather are inspired by developments in statistical physics in connection with correlation decay properties of Gibbs measures and its implications to uniqueness of Gibbs measures on infinite trees, reconstruction problems and local weak convergence methods. On a negative side, our algorithms provide $ε$-approximations only to the logarithms of the size of a feasible set (also known as free energy in statistical physics). But on the positive side, our approach provides deterministic as opposed to probabilistic guarantee on approximations. Moreover, for some regular graphs we obtain explicit values for the counting problem. For example, we show that every 4-regular $n$-node graph with large girth has approximately $(1.494...)^n$ independent sets, and in every $r$-regular graph with $n$ nodes and large girth the number of $q\geq r+1$-proper colorings is approximately $[q(1-{1\over q})^{r\over 2}]^n$, for large $n$. In statistical physics terminology, we compute explicitly the limit of the log-partition function. We extend our results to random regular graphs. Our explicit results would be hard to derive via the Markov chain method.

math.PR↗
arXiv2005

A survey of max-type recursive distributional equations

In certain problems in a variety of applied probability settings (from probabilistic analysis of algorithms to statistical physics), the central requirement is to solve a recursive distributional equation of the form X =^d g((ξ_i,X_i),i\geq 1). Here (ξ_i) and g(\cdot) are given and the X_i are independent copies of the unknown distribution X. We survey this area, emphasizing examples where the function g(\cdot) is essentially a ``maximum'' or ``minimum'' function. We draw attention to the theoretical question of endogeny: in the associated recursive tree process X_i, are the X_i measurable functions of the innovations process (ξ_i)?

math.PR↗