# Torsional rigidity in random walk spaces

## TL;DR

This paper investigates nonlocal torsional rigidity in random walk spaces, establishing its relation with spectral heat content, eigenvalues, and inequalities, thus extending classical results to nonlocal and graph settings.

## Contribution

It introduces a comprehensive framework connecting nonlocal torsional rigidity with spectral properties, inequalities, and graph structures, providing new insights and generalizations.

## Key findings

- Relation between nonlocal torsional rigidity and spectral m-heat content
- Recovery of the first eigenvalue via a limit formula
- Nonlocal Saint-Venant and Pólya-Makai inequalities

## Abstract

In this paper we study the (nonlocal) torsional rigidity in the ambient space of random walk spaces. We get the relation of the (nonlocal) torsional rigidity of a set $\Omega$ with the spectral $m$-heat content of $\Omega$, what gives rise to a complete description of the nonlocal torsional rigidity of $\Omega$ by using uniquely probability terms involving the set $\Omega$; and recover the first eigenvalue of the nonlocal Laplacian with homogeneous Dirichlet boundary conditions by a limit formula using these probability term. For the random walk in $\R^N$ associated with a non singular kernel, we get a nonlocal version of the Saint-Venant inequality, and, under rescaling we recover the classical Saint-Venant inequality. We study the nonlocal $p$-torsional rigidity and its relation with the nonlocal Cheeger constants. We also get a nonlocal version of the P\'{o}lya-Makai-type inequalities. We relate the torsional rigidity given here for weighted graphs with the torsional rigidity on metric graphs.