Transportation Distance between Probability Measures on the Infinite Regular Tree

SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 1113-1157, March 2024.
Abstract. In the infinite regular tree [math] with [math], we consider families [math], indexed by vertices [math] and nonnegative integers (“discrete time steps”) [math], of probability measures such that [math] if the distances [math] and [math] are equal. Let [math] be a positive integer, and let [math] and [math] be two vertices in the tree which are at distance [math] apart. We compute a formula for the transportation distance [math] in terms of generating functions. In the special case where [math] are measures from simple random walks after [math] time steps, we establish the linear asymptotic formula [math], as [math], and give the formulas for the coefficients [math] and [math] in closed forms. We also obtain linear asymptotic formulas when [math] is the uniform distribution on the sphere or on the ball of radius [math] as [math]. We show that these six coefficients (two from the simple random walk, two from the uniform distribution on the sphere, and two from the uniform distribution on the ball) are related by inequalities.