arXiv subjects

Moumanti Podder

Publications and source records attributed to Moumanti Podder.

29 records · Page 2Linked to original sources

arXiv2020

Avoidance couplings on non-complete graphs

A coupling of random walkers on the same finite graph, who take turns sequentially, is said to be an avoidance coupling if the walkers never collide. Previous studies of these processes have focused almost exclusively on complete graphs, in particular how many walkers an avoidance coupling can include. For other graphs, apart from special cases, it has been unsettled whether even two non-colliding simple random walkers can be coupled. In this article, we construct such a coupling on (i) any $d$-regular graph avoiding a fixed subgraph depending on $d$; and (ii) any square-free graph with minimum degree at least three. A corollary of the first result is that a uniformly random regular graph on $n$ vertices admits an avoidance coupling with high probability.

math.PR↗

arXiv2020

Uniform threshold for fixation of the stochastic sandpile model on the line

We consider the abelian stochastic sandpile model. In this model, a site is deemed unstable when it contains more than one particle. Each unstable site, independently, is toppled at rate $1$, sending two of its particles to neighbouring sites chosen independently. We show that when the initial average density is less than $1/2$, the system locally fixates almost surely. We achieve this bound by analysing the parity of the total number of times each site is visited by a large number of particles under the sandpile dynamics.

math.PR↗

arXiv2019

Random tree recursions: which fixed points correspond to tangible sets of trees?

Let $\mathcal{B}$ be the set of rooted trees containing an infinite binary subtree starting at the root. This set satisfies the metaproperty that a tree belongs to it if and only if its root has children $u$ and $v$ such that the subtrees rooted at $u$ and $v$ belong to it. Let $p$ be the probability that a Galton-Watson tree falls in $\mathcal{B}$. The metaproperty makes $p$ satisfy a fixed-point equation, which can have multiple solutions. One of these solutions is $p$, but what is the meaning of the others? In particular, are they probabilities of the Galton-Watson tree falling into other sets satisfying the same metaproperty? We create a framework for posing questions of this sort, and we classify solutions to fixed-point equations according to whether they admit probabilistic interpretations. Our proofs use spine decompositions of Galton-Watson trees and the analysis of Boolean functions.

math.PR↗

arXiv2019

Quantifier alternation in a class of recursively defined tree properties

Alternating quantifier depth is a natural measure of difficulty required to express first order logical sentences. We define a sequence of first order properties on rooted, locally finite trees in a recursive manner, and provide rigorous arguments for finding the alternating quantifier depth of each property in the sequence, using Ehrenfeucht-Fraïssé games.

math.LO↗

arXiv2018

First order theory on $G(n, c n^{-1})$

A well-known result of Shelah and Spencer tells us that the almost sure theory for first order language on the random graph sequence $\left\{G(n, cn^{-1})\right\}$ is not complete. This paper proposes and proves what the complete set of completions of the almost sure theory for $\left\{G(n, c n^{-1})\right\}$ should be. The almost sure theory $T$ consists of two sentence groups: the first states that all the components are trees or unicyclic components, and the second states that, given any $k \in \mathbb{N}$ and any finite tree $t$, there are at least $k$ components isomorphic to $t$. We define a $k$-completion of $T$ to be a first order property $A$, such that if $T + A$ holds for a graph, we can fully describe the first order sentences of quantifier depth $\leq k$ that hold for that graph. We show that a $k$-completion $A$ specifies the numbers, up to "cutoff" $k$, of the (finitely many) unicyclic component types of given parameters (that only depend on $k$) that the graph contains. A complete set of $k$-completions is then the finite collection of all possible $k$-completions.

math.PR↗

arXiv2017

Second order logic on random rooted trees

We address questions of logic and expressibility in the context of random rooted trees. Infiniteness of a rooted tree is not expressible as a first order sentence, but is expressible as an existential monadic second order sentence (EMSO). On the other hand, finiteness is not expressible as an EMSO. For a broad class of random tree models, including Galton-Watson trees with offspring distributions that have full support, we prove the stronger statement that finiteness does not agree up to a null set with any EMSO. We construct a finite tree and a non-null set of infinite trees that cannot be distinguished from each other by any EMSO of given parameters. This is proved via set-pebble Ehrenfeucht games (where an initial colouring round is followed by a given number of pebble rounds).

math.PR↗

arXiv2017

Gibbs measures over locally tree-like graphs and percolative entropy over infinite regular trees

Consider a statistical physical model on the $d$-regular infinite tree $T_{d}$ described by a set of interactions $Φ$. Let $\{G_{n}\}$ be a sequence of finite graphs with vertex sets $V_n$ that locally converge to $T_{d}$. From $Φ$ one can construct a sequence of corresponding models on the graphs $G_n$. Let $\{μ_n\}$ be the resulting Gibbs measures. Here we assume that $\{μ_{n}\}$ converges to some limiting Gibbs measure $μ$ on $T_{d}$ in the local weak$^*$ sense, and study the consequences of this convergence for the specific entropies $|V_n|^{-1}H(μ_n)$. We show that the limit supremum of $|V_n|^{-1}H(μ_n)$ is bounded above by the \emph{percolative entropy} $H_{perc}(μ)$, a function of $μ$ itself, and that $|V_n|^{-1}H(μ_n)$ actually converges to $H_{perc}(μ)$ in case $Φ$ exhibits strong spatial mixing on $T_d$. We discuss a few examples of well-known models for which the latter result holds in the high temperature regime.

math.PR↗

arXiv2016

First Order Probabilities For Galton-Watson Trees

In the regime of Galton-Watson trees, first order logic statements are roughly equivalent to examining the presence of specific finite subtrees. We consider the space of all trees with Poisson offspring distribution and show that such finite subtrees will be almost surely present when the tree is infinite. Introducing the notion of universal trees, we show that all first order sentences of quantifier depth $k$ depend only on local neighbourhoods of the root of sufficiently large radius depending on $k$. We compute the probabilities of these neighbourhoods conditioned on the tree being infinite. We give an almost sure theory for infinite trees.

math.PR↗

arXiv2016

Ruin probabilities under Sarmanov dependence structure

Our work aims to study the tail behaviour of weighted sums of the form $\sum_{i=1}^{\infty} X_{i} \prod_{j=1}^{i}Y_{j}$, where $(X_{i}, Y_{i})$ are independent and identically distributed, with common joint distribution bivariate Sarmanov. Such quantities naturally arise in financial risk models. Each $X_{i}$ has a regularly varying tail. With sufficient conditions similar to those used by Denisov and Zwart (2007) imposed on these two sequences, and with certain suitably summable bounds similar to those proposed by Hazra and Maulik (2012), we explore the tail distribution of the random variable $\sup_{n \geq 1}\sum_{i=1}^{n} X_i \prod_{j=1}^{i}Y_{j}$. The sufficient conditions used will relax the moment conditions on the $\{Y_{i}\}$ sequence.

math.PR↗

arXiv2016

Inverse Problems under Sarmanov dependence structure

Consider a sequence $\{(X_{i}, Y_{i})\}$ of independent and identically distributed random vectors, with joint distribution bivariate Sarmanov. This is a natural set-up for discrete time financial risk models with insurance risks. Of particular interest are the infinite time ruin probabilities $P\left[\sup_{n \geq 1}\sum_{i=1}^{n} X_i \prod_{j=1}^{i}Y_{j} > x\right]$. When the $Y_{i}$'s are assumed to have lighter tails than the $X_{i}$'s, we investigate sufficient conditions that ensure each $X_{i}$ has a regularly varying tail, given that the ruin probability is regularly varying. This is an inverse problem to the more traditional analysis of the ruin probabilities based on the tails of the $X_{i}$'s. We impose moment-conditions as well as non-vanishing Mellin transform assumptions on the $Y_{i}$'s in order to achieve the desired results. But our analysis departs from the more conventional assumption of independence between the sequences $\{X_{i}\}$ and $\{Y_{i}\}$, instead assuming each $(X_{i}, Y_{i})$ to be jointly distributed as bivariate Sarmanov, a fairly broad class of bivariate distributions.

math.PR↗

arXiv2016

Galton-Watson Probability Contraction

We are concerned with exploring the probabilities of first order statements for Galton-Watson trees with $Poisson(c)$ offspring distribution. Fixing a positive integer $k$, we exploit the $k$-move Ehrenfeucht game on rooted trees for this purpose. Let $Σ$, indexed by $1 \leq j \leq m$, denote the finite set of equivalence classes arising out of this game, and $D$ the set of all probability distributions over $Σ$. Let $x_{j}(c)$ denote the true probability of the class $j \in Σ$ under $Poisson(c)$ regime, and $\vec{x}(c)$ the true probability vector over all the equivalence classes. Then we are able to define a natural recursion function $Γ$, and a map $Ψ= Ψ_{c}: D \rightarrow D$ such that $\vec{x}(c)$ is a fixed point of $Ψ_{c}$, and starting with any distribution $\vec{x} \in D$, we converge to this fixed point via $Ψ$ because it is a contraction. We show this both for $c \leq 1$ and $c > 1$, though the techniques for these two ranges are quite different.

math.PR↗