arXiv subjects

Moumanti Podder

Publications and source records attributed to Moumanti Podder.

At least 19 recordsLinked to original sources

arXiv2026

Elephant random walk with attributed steps and extractions of random sizes

We study a model of market economics wherein the $(n+1)$-st customer, for each $n\geqslant N$, with $N$ being a prespecified positive integer, draws a sample of (random) size $K_{n}$, either with replacement or without, from the customers of the past. Each sampled customer is queried as to which of the two products, A and B, available in the oligopolistic market, they chose, and whether they are satisfied or not with their choice. The $(n+1)$-st customer now employs a stochastic rule, based on the information collected from the sampled customers, to decide which of the two products to buy. The probability that a customer is satisfied with the product they have purchased equals $q_{1}$ when the product is A, and $q_{2}$ when it is B, independent of all else. The resulting stochastic process may be represented as a variant of the celebrated elephant random walk, with the relative performance (in terms of sale) of A with respect to B, up to and including the $n$-th sale, captured by the position $S_{n}$ of the walker at time $n$. We study the almost sure convergence of $S_{n}/n$, as well as the convergence in distribution of suitably scaled versions of $S_{n}$ (where the scaling depends on the regime we are in).

math.PR↗

arXiv2026

An exactly solvable evaporation-deposition PCA with long-distance interactions

We consider a probabilistic cellular automaton (PCA) of evaporation-deposition on the one-dimensional lattice having $n$ sites with periodic boundary conditions, in which each site, during each epoch, can be in one of two states: $0$ and $1$. Fix a positive integer $m\geqslant 2$. There are two types of transitions at each discrete time, which are as follows: (i) the first site in every contiguous block of $m$ $0$s becomes a $1$ with probability $p_1$, and (ii) the first site in every contiguous block of $(m-1)$ $0$s followed immediately by a $1$ also becomes a $1$ with probability $(1-p_2)$. As in a PCA, all of these transitions occur simultaneously. We show that the resulting discrete-time Markov chain is ergodic, and we give an explicit formula for its limiting distribution, the partition function and the density. We also propose necessary and sufficient conditions for this Markov chain to be reversible. For $m=2$, we provide a fully analytical expression for the free energy of this model.

math.PR↗

arXiv2025

The Maker-Breaker directed triangle game

In this work, we investigate Maker-Breaker directed triangle games, a directionally constrained variant of the classical Maker-Breaker triangle game. Our board of interest is a tournament, and the winning sets are all $3$-cycles present in the tournament. We begin by studying the Maker-Breaker directed triangle game played on a specially defined tournament called the parity tournament, and we identify the board size threshold to be $n=7$, which is to say that for a parity tournament on $n$ vertices, Breaker has a winning strategy for $3\le n<7$, while Maker can ensure a win for herself for $n\ge7$. For the $(1:b)$ biased version of this game, we prove that the bias threshold $b^*(n)$ satisfies $\sqrt{\left(1/12+o(1)\right)n}\le b^*(n)\le\sqrt{\left(8/3+o(1)\right)n}$, which matches the order of magnitude, namely $\sqrt n$, of the bias threshold for the undirected counterpart of this game. Next, we consider the game on random tournaments $T(n,p)$, wherein the edge between $i$ and $j$, for each $i<j$, is directed from $i$ to $j$ with probability $p$, independent of all else. We prove that Maker wins this game with probability tending to $1$ as $n\to\infty$ for any fixed $p\in(0,1)$. Extending the notion of bias from undirected games to our directed framework, we introduce the flip-biased Maker-Breaker directed triangle game on the parity tournament with flip budget $\kappa(n)$, where Breaker may strategically flip the directions of at most $\kappa(n)$ edges before the game begins. We show that the flip-bias threshold $\kappa^*(n)$ for this game is of order $n^2$. More precisely, for odd $n\ge11$ we show $n(n-11)/12\le\kappa^*(n)\le (n^2-1)/8$, and for even $n\ge14$ we show $n(n-14)/12+1\le\kappa^*(n)\le n^2/8+n/4-1$.

math.CO↗

arXiv2025

The Game of Graph Nim on graphs with four edges

This work is concerned with the study of the Game of Graph Nim -- a class of two-player combinatorial games -- on graphs with $4$ edges. To each edge of such a graph is assigned a positive-integer-valued edge-weight, and during each round of the game, the player whose turn it is to make a move selects a vertex, and removes a non-negative integer edge-weight from each of the edges incident on that vertex, such that the remaining edge-weight on each of these edges is a non-negative integer, and the total edge-weight removed during a round is strictly positive. The game continues for as long as the sum of the edge-weights remaining on all edges of the graph is strictly positive, and the player who plays the last round wins. An initial configuration of edge-weights is considered winning if the player who plays the first round wins the game, whereas it is defined as losing if the player who plays the second round wins. In this paper, we characterize, almost entirely, all winning and losing configurations for this game on all graphs with precisely $4$ edges each. Only one such graph defies our attempt to fully characterize the winning and losing configurations of edge-weights on its edges -- we are still able to provide a significant set of partial results pertaining to this graph.

math.CO↗

arXiv2025

Elephant random walks with multiple extractions and general reinforcement functions

We consider a generalized model of elephant random walks wherein the walker, during the $(n+1)$-st time-stamp, draws from the past (i.e. the set $\{1,2,\ldots,n\}$) a sample of $k$ time-stamps, either with replacement or without, where $k$ may either remain fixed as $n$ grows, or $k=k(n)$ may grow with $n$. Letting $\{U_{n,1}, U_{n,2}, \ldots, U_{n,k}\}$ denote the time-stamps sampled, the step taken by the walker during the $(n+1)$-st time-stamp, denoted $X_{n+1}$, is a $\pm 1$-valued random variable whose distribution depends on the proportion of $(+1)$-valued steps out of $X_{U_{n,1}},X_{U_{n,2}},\ldots,X_{U_{n,k}}$ via a reinforcement function $f$. In this paper, we investigate the asymptotic behaviour, i.e. strong and weak convergence, of this random walk model under suitable assumptions made on the function $f$ (as well as on the sequence $\{k(n)\}$ when the sample size varies with $n$).

math.PR↗

arXiv2024

Nonparametric method of structural break detection in stochastic time series regression model

We propose a novel nonparametric test to detect structural breaks in the conditional mean and/or variance of a time series. Our method does not assume any specific parametric form for the dependence structure of the regressor, the time series model, or the distribution of the noise. This flexibility allows our algorithm to be applicable to a wide range of framework. We further apply the proposed test to accurately localize the changepoints and establish theoretical guarantees showing that the estimated structural breaks are consistent, meaning they lie sufficiently close to the true breakpoints when a sufficiently large sample is available. The effectiveness of the proposed algorithm is demonstrated through an extensive simulation study encompassing a diverse range of time series structures, including light, moderately heavy, and heavy tailed distributions. We also show a real-life example, where an application to Bitcoin prices and Google search volume illustrates how the procedure can identify changes in the conditional relationship between market attention and price dynamics.

stat.ME↗

arXiv2024

Largest eigenvalue of positive mean Gaussian matrices

This short note studies the fluctuations of the largest eigenvalue of symmetric random matrices with correlated Gaussian entries having positive mean. Under the assumption that the covariance kernel is absolutely summable, it is proved that the largest eigenvalue, after centering, converges in distribution to normal with an explicitly defined mean and variance. This result generalizes known findings for Wigner matrices with independent entries.

math.PR↗

arXiv2024

Percolation games on rooted, edge-weighted random trees

Consider a rooted Galton-Watson tree $T$, to each of whose edges we assign, independently, a weight that equals $+1$ with probability $p_{1}$, $0$ with probability $p_{0}$ and $-1$ with probability $p_{-1}=1-p_{1}-p_{0}$. We play a game on a realization of this tree, involving two players and a token that is allowed to be moved from where it is currently located, say a vertex $u$ of $T$, to any child $v$ of $u$. The players begin with initial capitals that amount to $i$ and $j$ units respectively, and a player wins if either she is the first to amass a capital worth $\kappa$ units, where $\kappa \in \mathbb{N}$ is prespecified, or she is able to move the token to a leaf vertex, from where her opponent cannot move it any farther, or her opponent's capital is the first to dwindle to $0$. This paper is concerned with analyzing the probabilities of the three possible outcomes such a game may culminate in, as well as with finding conditions under which the expected duration of the game is finite. Of particular interest to us is the exploration of criteria that guarantee the probability of draw in such a game to be $0$. The theory we develop is further supported by observations obtained via computer simulations, providing a deeper insight into how the above-mentioned probabilities behave as the underlying parameters and / or offspring distributions are allowed to vary. We include in this paper conjectures pertaining to the behaviour of the probability of draw in our game (including a phase transition phenomenon, in which the probability of draw goes from being $0$ to being strictly positive) as the parameter-pair $(p_{0},p_{1})$ is varied suitably while keeping the underlying offspring distribution of $T$ fixed.

math.PR↗

arXiv2024

Generalized percolation games on the $2$-dimensional square lattice, and ergodicity of associated probabilistic cellular automata

Each vertex of the infinite $2$-dimensional square lattice graph is assigned, independently, a label that reads trap with probability $p$, target with probability $q$, and open with probability $(1-p-q)$, and each edge is assigned, independently, a label that reads trap with probability $r$ and open with probability $(1-r)$. A percolation game is played on this random board, wherein two players take turns to make moves, where a move involves relocating the token from where it is currently located, say $(x,y) \in \mathbb{Z}^{2}$, to one of $(x+1,y)$ and $(x,y+1)$. A player wins if she is able to move the token to a vertex labeled a target, or force her opponent to either move the token to a vertex labeled a trap or along an edge labeled a trap. We seek to find a regime, in terms of $p$, $q$ and $r$, in which the probability of this game resulting in a draw equals $0$. We consider special cases of this game, such as when each edge is assigned, independently, a label that reads trap with probability $r$, target with probability $s$, and open with probability $(1-r-s)$, but the vertices are left unlabeled. Various regimes of values of $r$ and $s$ are explored in which the probability of draw is guaranteed to be $0$. We show that the probability of draw in each such game equals $0$ if and only if a certain probabilistic cellular automaton (PCA) is ergodic, following which we implement the technique of weight functions to investigate the regimes in which said PCA is ergodic.

math.PR↗

arXiv2024

Learning models on rooted regular trees with majority update policy: convergence and phase transition

We study a learning model in which an agent is stationed at each vertex of $\mathbb{T}_{m}$, the rooted tree in which each vertex has $m$ children. At any time-step $t \in \mathbb{N}_{0}$, they are allowed to select one of two available technologies: $B$ and $R$. Let the technology chosen by the agent at vertex $v\in\mathbb{T}_{m}$, at time-step $t$, be $C_{t}(v)$. Let $\{C_{0}(v):v\in\mathbb{T}_{m}\}$ be i.i.d., where $C_{0}(v)=B$ with probability $\pi_{0}$. During epoch $t$, the agent at $v$ performs an experiment that results in success with probability $p_{B}$ if $C_{t}(v)=B$, and with probability $p_{R}$ if $C_{t}(v)=R$. If the children of $v$ are $v_{1},\ldots,v_{m}$, the agent at $v$ updates their technology to $C_{t+1}(v)=B$ if the number of successes among all $v_{i}$ with $C_{t}(v_{i})=B$ exceeds, strictly, the number of successes among all $v_{j}$ with $C_{t}(v_{j})=R$. If these numbers are equal, then the agent at $v$ sets $C_{t+1}(v)=B$ with probability $1/2$. Else, $C_{t+1}(v)=R$. We show that $\{C_{t}(v):v\in\mathbb{T}_{m}\}$ is i.i.d., where $C_{t}(v)=B$ with probability $\pi_{t}$, and $\{\pi_{t}\}_{t \in \mathbb{N}_{0}}$ converges to a fixed point $\pi$ of a function $g_{m}$. For $m \geqslant 3$, there exists a $p(m) \in (0,1)$ such that $g_{m}$ has a unique fixed point, $1/2$, when $p \leqslant p(m)$, and three distinct fixed points, of the form $\alpha$, $1/2$ and $1-\alpha$, when $p > p(m)$. When $m=3$, $p_{B}=1$ and $p_{R} \in [0,1)$, we show that $g_{3}$ has a unique fixed point, $1$, when $p_{R} < \sqrt{3}-1$, two distinct fixed points, one of which is $1$, when $p_{R} = \sqrt{3}-1$, and three distinct fixed points, one of which is $1$, when $p_{R} > \sqrt{3}-1$. When $g_{m}$ has multiple fixed points, we also specify which of these fixed points $\pi$ equals, depending on $\pi_{0}$. For $m=2$, we describe the behaviour of $g_{3}$ for all $p_{B}$ and $p_{R}$.

math.PR↗

arXiv2023

Phase transition in percolation games on rooted Galton-Watson trees

We study the bond percolation game and the site percolation game on the rooted Galton-Watson tree $T_{\chi}$ with offspring distribution $\chi$. We obtain the probabilities of win, loss and draw for each player in terms of the fixed points of functions that involve the probability generating function $G$ of $\chi$, and the parameters $p$ and $q$. Here, $p$ is the probability with which each edge (respectively vertex) of $T_{\chi}$ is labeled a trap in the bond (respectively site) percolation game, and $q$ is the probability with which each edge (respectively vertex) of $T_{\chi}$ is labeled a target in the bond (respectively site) percolation game. We obtain a necessary and sufficient condition for the probability of draw to be $0$ in each game, and we examine how this condition simplifies to yield very precise phase transition results when $\chi$ is Binomial$(d,\pi)$, Poisson$(\lambda)$, or Negative Binomial$(r,\pi)$, or when $\chi$ is supported on $\{0,d\}$ for some $d \in \mathbb{N}$, $d \geqslant 2$. It is fascinating to note that, while all other specific classes of offspring distributions we consider in this paper exhibit phase transition phenomena as the parameter-pair $(p,q)$ varies, the probability that the bond percolation game results in a draw remains $0$ for all values of $(p,q)$ when $\chi$ is Geometric$(\pi)$, for all $0 < \pi \leqslant 1$. By establishing a connection between these games and certain finite state probabilistic tree automata on rooted $d$-regular trees, we obtain a precise description of the regime (in terms of $p$, $q$ and $d$) in which these automata exhibit ergodicity or weak spatial mixing.

math.PR↗

arXiv2023

The spread of an epidemic: a game-theoretic approach

We introduce and study a model stemming from game theory for the spread of an epidemic throughout a given population. Each agent is allowed to choose an action whose value dictates to what extent they limit their social interactions, if at all. Each of them is endowed with a certain amount of immunity such that if the viral risk/exposure is more than that they get infected. We consider a discrete-time stochastic process where, at the beginning of each epoch, a randomly chosen agent is allowed to update their action, which they do with the aim of maximizing a utility function that is a function of the state in which the process is currently in. The state itself is determined by the subset of infected agents at the beginning of that epoch, and the most recent action profile of all the agents. Our main results are concerned with the limiting distributions of both the cardinality of the subset of infected agents and the action profile as time approaches infinity, considered under various settings (such as the initial action profile we begin with, the value of each agent's immunity etc.). We also provide some simulations to show that the final asymptotic distributions for the cardinality of infected set are almost always achieved within the first few epochs.

math.PR↗

arXiv2022

Ergodicity of a generalized probabilistic cellular automaton with parity-based neighbourhoods

We study a one-dimensional generalized probabilistic cellular automaton $E_{p, q}$ with universe $\mathbb Z$, alphabet $\mathcal A = \{0, 1\}$, parameters $p$ and $q$ such that $0 < p+q \leq 1$ and two neighbourhoods $\mathcal N_0 = \{0, 1\}$ and $\mathcal N = \{1, 2\}$. The state $E_{p, q} \eta (x)$ of any $x \in \mathbb Z$ under the application of $E_{p, q}$ is a random variable whose probability distribution depends on the states $\eta(x + y)$ for $y \in \mathcal N_i$ where $i$ has the same parity as $x$. We establish ergodicity of this GPCA for various ranges of values of $p$ and $q$ via its connection with a suitable percolation game on a two-dimensional lattice. For these same ranges of values of $p$ and $q$, we show that the above-mentioned game has probability $0$ of resulting in a draw.

math.PR↗

arXiv2022

On a class of probabilistic cellular automata with size-$3$ neighbourhood and their applications in percolation games

Different versions of percolation games on $\mathbb{Z}^{2}$, with parameters $p$ and $q$ that indicate, respectively, the probability with which a site in $\mathbb{Z}^{2}$ is labeled a trap and the probability with which it is labeled a target, are shown to have probability $0$ of culminating in draws when $p+q > 0$. We show that, for fixed $p$ and $q$, the probability of draw in each of these games is $0$ if and only if a certain $1$-dimensional probabilistic cellular automaton (PCA) $F_{p,q}$ with a size-$3$ neighbourhood is ergodic. This allows us to conclude that $F_{p,q}$ is ergodic whenever $p+q > 0$, thereby rigorously establishing ergodicity for a considerable class of PCAs.

math.PR↗

arXiv2022

Combinatorial games on Galton-Watson trees involving several-generation-jump moves

We study the $k$-jump normal and $k$-jump mis\`{e}re games on rooted Galton-Watson trees, expressing the probabilities of various outcomes of these games as specific fixed points of certain functions that depend on $k$ and the offspring distribution. We discuss results on phase transitions pertaining to draw probabilities when the offspring distribution is Poisson$(\lambda)$ (i.e. for which values of $\lambda$, the draw probability is strictly positive). We compare the probabilities of the various outcomes of the $2$-jump normal game with those of the $2$-jump mis\`{e}re game, and a similar comparison is drawn between the $2$-jump normal game and the $1$-jump normal game, under the Poisson regime. We describe the rate of decay of the probability that the first player loses the $2$-jump normal game as $\lambda \rightarrow \infty$. Finally, we discuss a sufficient condition for the average duration of the $k$-jump normal game to be finite.

math.PR↗

arXiv2021

Combinatorial games on multi-type Galton-Watson trees

When normal and misère games are played on bi-type binary Galton-Watson trees (with vertices coloured blue or red and each having either no child or precisely $2$ children), with one player allowed to move along monochromatic edges and the other along non-monochromatic edges, the draw probabilities equal $0$ unless every vertex gives birth to one blue and one red child. On bi-type Poisson trees where each vertex gives birth to Poisson$(λ)$ offspring in total, the draw probabilities approach $1$ as $λ\rightarrow \infty$. We study such \emph{novel} versions of normal, misère and escape games on rooted multi-type Galton-Watson trees, with the "permissible" edges for one player being disjoint from those of her opponent. The probabilities of the games' outcomes are analyzed, compared with each other, and their behaviours as functions of the underlying law explored.

math.PR↗

arXiv2021

Uniqueness of communities in regular stochastic block models

This paper studies the regular stochastic block model comprising \emph{several} communities: each of the $k$ non-overlapping communities, for $k \geqslant 3$, possesses $n$ vertices, each of which has total degree $d$. The values of the intra-cluster degrees (i.e.\ the number of neighbours of a vertex inside the cluster it belongs to) and the inter-cluster degrees (i.e.\ the number of neighbours of a vertex inside a cluster different from its own) are allowed to vary across clusters. We discuss two main results: the first compares the probability measure induced by our model with the uniform measure on the space of $d$-regular graphs on $kn$ vertices, and the second establishes that the clusters, under rather weak assumptions, are unique asymptotically almost surely as $n \rightarrow \infty$.

math.PR↗

arXiv2020

Zero-one laws for existential first order sentences of bounded quantifier depth

For any fixed positive integer $k$, let $α_{k}$ denote the smallest $α\in (0,1)$ such that the random graph sequence $\left\{G\left(n, n^{-α}\right)\right\}$ does not satisfy the zero-one law for the set $\mathcal{E}_{k}$ of all existential first order sentences that are of quantifier depth at most $k$. This paper finds upper and lower bounds on $α_{k}$, showing that as $k \rightarrow \infty$, we have $α_{k} = \left(k - 2 - t(k)\right)^{-1}$ for some function $t(k) = Θ(k^{-2})$. We also establish the precise value of $α_{k}$ when $k = 4$.

math.PR↗