The Mézard-Parisi equation for matchings in pseudo-dimension $d>1$

Abstract

We establish existence and uniqueness of the solution to the cavity equation for the random assignment problem in pseudo-dimension d > 1, as conjectured by Aldous and Bandyopadhyay (Annals of Applied Probability, 2005) and Wästlund (Annals of Mathematics, 2012). This fills the last remaining gap in the proof of the original Mézard-Parisi prediction for this problem (Journal de Physique Lettres, 1985).Keywords: recursive distributional equation; random assignment problem; mean-field combinatorial optimization; cavity method.2010 MSC: 60C05, 82B44, 90C35.

Cut-off method for endogeny of recursive tree processes

“…Therefore, from now on, we shall only consider measures satisfying (35) and actually after four iterations we can restrict even more the space on which we study the behaviour of Φ. Indeed, using the asymptotic equivalences (33) and (34), proceeding as in [7, Lemma 3] and [16,Prop. 3.1], we have:…”

“…To overpass this problematic issue, the solution is to restrict to a space Q of measures whose tails have a sufficiently good integrability condition, as done first in [17] and then in [7,16]. Indeed, note that from the expression (31) we have the asymptotic equivalences…”

“…6.5 Solutions to the equations -Actually, a cut-off method has already been used in [11,16,19] and [7] to solve the RDEs (29) and (30) respectively. Combining these works one can state:…”

“…2], the statement of Theorem 2 holds also provided that the measure m is absolutely continuous with everywhere positive continuous density and has normal tails at ±∞. Interestingly, different approaches by cut-off have been used to find solutions to the RDEs that are associated with problems of mean-field combinatorial optimization (see Section 6, as well as [2], [3, § 7]): the mean-field approximation for minimal matching in pseudo-dimension 0 < q < ∞ [11,16,19], and for the minimum weight k-factor problem, which corresponds to the mean-field travelling salesman problem when k = 2 [15,18]. For the latter, we point out that [7] makes no use of cut-off to show existence of the solution, although many ideas are in the spirit of [9], when proving that the solution is a global attractor.…”

“…Relying on these works, we apply our main result to show the endogeny. For the minimal matching, the case q = 1 was already solved in [4]; here we use the results in [16,19] to treat the problem of pseudo-dimension q > 1. This completes Aldous' program for the rigorous analysis of physical predictions for these models (see [3, § 7.5] and [8,17]).…”