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Partha Dey

Publications and source records attributed to Partha Dey.

6 recordsLinked to original sources
arXiv2024

Network evolution with mesoscopic delay

Owing to the influence of real-world networks both in science and society, numerous mathematical models have been developed to understand the structure and evolution of these systems, particularly in a temporal context. Recent advancements in fields like distributed cyber-security and social networks have spurred the creation of probabilistic models of evolution, where individuals make decisions based on only partial information about the network's current state. This paper seeks to explore models incorporating network delay, where new participants receive information from a time-lagged snapshot of the system. In the context of mesoscopic network delays, we develop probabilistic tools built on stochastic approximation to understand asymptotics of both local functionals, such as local neighborhoods and degree distributions, as well as global properties, such as the evolution of the degree of the network's initial founder. A companion paper explores the regime of macroscopic delays in the evolution of the network.

math.PR↗
arXiv2024

Network evolution with Macroscopic Delays: asymptotics and condensation

Preferential attachment models typically assume that each arriving vertex observes the current network before choosing its connection. Motivated by distributed systems and social networks, we study network delay, where this decision uses only a time-delayed snapshot. We focus on macroscopic delays, for which the delay is proportional to the current network size and hence removes a non-vanishing fraction of the available information. We identify the local weak limit as a continuous-time branching process whose reproduction point process has memory of its entire past. Since this non-Markovian description is difficult to analyze directly, we construct a dual branching process in which edges reproduce, recovering enough independence for quantitative analysis. This yields a detailed understanding of how the delay affects features such as the tail behavior of the asymptotic degree distribution, together with necessary and sufficient conditions for condensation-the phenomenon in which a positive fraction of the degree mass escapes to infinity. We conclude by studying the impact of the delay distribution on macroscopic functionals such as the root degree.

math.PR↗
arXiv2021

Fluctuation results for size of the vacant set for random walks on discrete torus

We consider one or more independent random walks on the $d\ge 3$ dimensional discrete torus. The walks start from vertices chosen independently and uniformly at random. We analyze the fluctuation behavior of the size of some random sets arising from the trajectories of the random walks at a time proportional to the size of the torus. Examples include vacant sets and the intersection of ranges. The proof relies on a refined analysis of tail estimates for hitting time and can be applied for other vertex-transitive graphs.

math.PR↗
arXiv2018

Longest increasing path within the critical strip

A Poisson point process of unit intensity is placed in the square $[0,n]^2$. An increasing path is a curve connecting $(0,0)$ with $(n,n)$ which is non-decreasing in each coordinate. Its length is the number of points of the Poisson process which it passes through. Baik, Deift and Johansson proved that the maximal length of an increasing path has expectation $2n-n^{1/3}(c_1+o(1))$, variance $n^{2/3}(c_2+o(1))$ and that it converges to the Tracy-Widom distribution after suitable scaling. Johansson further showed that all maximal paths have a displacement of $n^{\frac23+o(1)}$ from the diagonal with probability tending to one as $n\to \infty$. Here we prove that the maximal length of an increasing path restricted to lie within a strip of width $n^γ, γ<\frac23$, around the diagonal has expectation $2n-n^{1-γ+o(1)}$, variance $n^{1 - \fracγ{2}+o(1)}$ and that it converges to the Gaussian distribution after suitable scaling.

math.PR↗
arXiv2015

Fluctuations of the free energy in the mixed $p$-spin models with external field

We show that the free energy in the mixed $p$-spin models of spin glasses does not superconcentrate in the presence of external field, which means that its variance is of the order suggested by the Poincaré inequality. This complements the result of Chatterjee who showed that the free energy superconcentrates when there is no external field. For models without odd $p$-spin interactions for $p\geq 3$, we prove the central limit theorem for the free energy at any temperature and give an explicit formula for the limiting variance. Although we only deal with the case of Ising spins, all our results can be extended to the spherical models as well.

math.PR↗