Microscopic sorting in a multispecies exclusion process

Balázs Maga; Bálint Tóth

Home/Archive/arXiv

arXiv · 2610.08135

Microscopic sorting in a multispecies exclusion process

Balázs Maga·Bálint Tóth

View original source ↗Read source document ↗Export metadata ↓

Abstract

We study a one-dimensional multispecies TASEP model with each particle type having an associated speed parameter and faster particles overtaking slower ones with the rate being the positive part of their speed difference. After expressing the sorting time of a finite system given a reverse-speed initial condition as the maximum of sorting times for all different fast-slow cuts, we prove that forbidding fast-fast and slow-slow overtakes does not alter the law of the sorting time. This allows for an LPP formulation and an analytic expression of the sorting time.

Explore related subjects

math.PR

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Balázs Maga, Bálint Tóth. 2026-10-06. Microscopic sorting in a multispecies exclusion process. https://arxiv.org/abs/2610.08135

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Highest Trees of Random Mappings

We study the heights of the trees of a random mapping of $n$ elements. Using singularity analysis of exact generating functions, we prove that the mapping has a unique highest tree with probability $1 - \sqrt{\fracπ{8}}\, n^{-1/2} + o(n^{-1/2})$. The property of having a unique highest tree plays a crucial role in the solution of the Road Coloring Problem [Trahtman, 2009]. More generally, we consider $c$-branches, the subtrees rooted at distance $c$ from the cycles. For all fixed $c \ge 0$ and $j \ge 1$ we show that the highest $c$-branch exceeds every other $c$-branch in height by at least $j$, except with probability asymptotic to $(2j-1) \sqrt{\fracπ{8}}\, n^{-1/2}$. We also show that, for any fixed $α> 1$, with probability $1 - Θ(n^{-1/2})$ the part of the highest $c$-branch that lies above the second highest one (its crown) has more than $α$ times as many vertices as roots. The last result is used in the author's proof that a random $2$-letter automaton with $n$ states is synchronizing with probability $1 - Θ(1/n)$.

math.PR↗

arXiv2026

Rapid mixing in unimodal landscapes and efficient simulatedannealing for multimodal distributions

We consider nearest neighbor weighted random walks on the $d$-dimensional box $[n]^d$ that are governed by some function $g:[0,1] \ra [0,\iy)$, by which we mean that standing at $x$, a neighbor $y$ of $x$ is picked at random and the walk then moves there with probability $(1/2)g(n^{-1}y)/(g(n^{-1}y)+g(n^{-1}x))$. We do this for $g$ of the form $f^{m_n}$ for some function $f$ which assumed to be analytically well-behaved and where $m_n \ra \iy$ as $n \ra \iy$. This class of walks covers an abundance of interesting special cases, e.g., the mean-field Potts model, posterior collapsed Gibbs sampling for Latent Dirichlet allocation and certain Bayesian posteriors for models in nuclear physics. The following are among the results of this paper: \begin{itemize} \item If $f$ is unimodal with negative definite Hessian at its global maximum, then the mixing time of the random walk is $O(n\log n)$. \item If $f$ is multimodal, then the mixing time is exponential in $n$, but we show that there is a simulated annealing scheme governed by $f^K$ for an increasing sequence of $K$ that mixes in time $O(n^2)$. Using a varying step size that decreases with $K$, this can be taken down to $O(n\log n)$. \item If the process is studied on a general graph rather than the $d$-dimensional box, a simulated annealing scheme expressed in terms of conductances of the underlying network, works similarly. \end{itemize} Several examples are given, including the ones mentioned above.

math.PR↗

arXiv2026

Picard Approximation and Martingale Integrand Estimation for Path-Dependent FBSDEs

This paper develops a Picard iteration method for path-dependent forward-backward stochastic differential equations (FBSDEs). We construct a Picard iteration scheme, prove its convergence to the exact solution, and derive an explicit convergence rate. Although the underlying problem is conceptually straightforward, establishing rigorous convergence results in the path-dependent setting is technically demanding. Path dependence introduces a logarithmic correction: the time-discretization error bound is of order $|π|\log(1/|π|)$, compared with the order $|π|$ obtained in the Markovian setting, where $|π|$ denotes the mesh size of the time partition. We also introduce an estimator for the martingale integrand of the FBSDE that is tailored to path dependence and establish, within a Monte Carlo framework, a concentration inequality that quantifies its statistical error. The applicability of the proposed method is demonstrated in two non-Markovian derivative-pricing settings. The primary objective of this paper is to provide a theoretical convergence analysis of the Picard iteration method in the path-dependent setting and illustrate its application to option pricing, rather than to develop a faster or more computationally efficient algorithm.

math.PR↗