Transportation Distance between Probability Measures on the Infinite Regular Tree

TL;DR

This paper derives formulas for transportation distances between probability measures on an infinite regular tree, including asymptotic behaviors for random walks and geometric measures, revealing relationships among key coefficients.

Contribution

It provides explicit formulas for transportation distances on infinite regular trees and establishes asymptotic linear formulas for various probability measures, connecting these results through inequalities.

Findings

Explicit formula for $W_1$ between measures at vertices

Linear asymptotics for random walk measures as time grows

Relationships and inequalities among coefficients for different measures

Abstract

In the infinite regular tree $\mathbb{T}_{q+1}$ with $q \in \mathbb{Z}_{\ge 2}$, we consider families $\{\mu_u^n\}$, indexed by vertices $u$ and nonnegative integers ("discrete time steps") $n$, of probability measures such that $\mu_u^n(v) = \mu_{u'}^n(v')$ if the distances $\operatorname{dist}(u,v)$ and $\operatorname{dist}(u',v')$ are equal. Let $d$ be a positive integer, and let $X$ and $Y$ be two vertices in the tree which are at distance $d$ apart. We compute a formula for the transportation distance $W_1\!\left( \mu_X^n, \mu_Y^n \right)$ in terms of generating functions. In the special case where $\mu_u^n = \mathfrak{m}_u^n$ are measures from simple random walks after $n$ time steps, we establish the linear asymptotic formula $W_1\!\left( \mathfrak{m}_X^n, \mathfrak{m}_Y^n \right) = An + B + o(1)$, as $n \to \infty$, and give the formulas for the coefficients $A$ and $B$ in closed forms. We also obtain linear asymptotic formulas in the cases of spheres and uniform balls as the radii tend to infinity. We show that these six coefficients (two from simple random walks, two from spheres, and two from uniform balls) are related by inequalities.