Scaling hypothesis for the Euclidean bipartite matching problem

Abstract

We propose a simple yet very predictive form, based on a Poisson's equation, for the functional dependence of the cost from the density of points in the Euclidean bipartite matching problem. This leads, for quadratic costs, to the analytic prediction of the large N limit of the average cost in dimension d = 1,2 and of the subleading correction in higher dimension. A nontrivial scaling exponent, γ(d) = d-2/d, which differs from the monopartite's one, is found for the subleading correction. We argue that the same scaling holds true for a generic cost exponent in dimension d > 2.

Scaling hypothesis for the Euclidean bipartite matching problem. II. Correlation functions

“…The previous results for the d = 1 case are known to the literature [7,8,11], although the correlation function was derived using a different probabilistic approach. In Ref.…”

“…Observe now that C d (0) is, in the large N limit, the average optimal cost for the Euclidean bipartite matching problem; using this simple correspondence, Caracciolo et al [11] derived the correct scaling of the optimal cost and, through a proper regularization procedure, the finite-size corrections to the average optimal cost for any dimension. In the following we will consider the complete correlation function C d (x) in any dimension and we will derive it using the same ansatz successfully adopted by Caracciolo et al [11] to obtain the scaling of the average optimal cost.…”

“…Analyzing the scaling of the total cost and performing a proper regularization of the previous quantity, Caracciolo et al [11] obtained…”

“…In the following we will consider the complete correlation function C d (x) in any dimension and we will derive it using the same ansatz successfully adopted by Caracciolo et al [11] to obtain the scaling of the average optimal cost. We will distinguish two different cases.…”

“…To introduce our results, in Sec. II we will review the Monge-Kantorovič formulation of the optimal transport problem, from which a suitable general ansatz, already used by Caracciolo et al [11], is derived for the expression of the optimal matching ray in the continuum limit for the p = 2 case. Using this working ansatz, in Sec.…”