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.

We propose a new approach for the study of the quadratic stochastic Euclidean bipartite matching problem between two sets of N points each, N 1. The points are supposed independently randomly generated on a domain Ω ⊂ R d with a given distribution ρ(x) on Ω. In particular, we derive a general expression for the correlation function and for the average optimal cost of the optimal matching. A previous ansatz for the matching problem on the flat hypertorus is obtained as particular case.The Euclidean bipartite matching problem (Ebmp) was firstly introduced and studied by Monge [1] in 1781. It is an assignment problem in which an underlying geometric structure is present. Assignment problems are of paramount importance in theoretical computer science [2,3] and a polynomial-time algorithm, the celebrated Hungarian algorithm [4][5][6], is available for their solution. In the Ebmp two sets of N points, let us call themThe problem, in its quadratic version, asks for the permutation π ∈ S N , S N symmetric group of N elements, such that the cost functionalis minimized. In the previous formula we have introducedand we have denoted by • the Euclidean norm in R d . Matching problems appear in many different physical, biological and computational applications. The (linear) Ebmp, for example, was introduced by Monge to optimize the transport cost of soil from N mining sites to N construction sites. The problem of covering a given lattice with dimers can also be reformulated as a matching problem [7], whereas, in computational biology, matching techniques are applied to pattern recognition problems [8]. In computer vision, the quadratic Ebmp is at the basis of many image stitching and stereographic reconstruction algorithms [9]. Finally, the quadratic cost functional in Eq. (1) plays a special role in physical applications. Indeed, it was used by Tanaka [10] in the study of Boltzmann equation, and by Brenier [11] in his variational formulation of Euler incompressible fluids.In many applications, however, the parameters (for example, the positions of the points) are affected by uncertainty, and the matching problem is a stochastic (or random) optimization problem. In this case, the average properties of the solution are of some interest.Many analytical and numerical techniques, derived from statistical physics [12,13], were successfully applied to the study of stochastic optimization problems. In particular, in the random assignment problem (rap), the quantities µ j (i) are considered independent and identically distributed random variables and the Euclidean structure is completely neglected. The rap was one of the first stochastic optimization problems to be solved using the theory of disordered systems by Mézard and Parisi [14]. Their results, obtained for N → ∞, were rigorously derived years later by Aldous [15]. Subsequently, Linusson and Wästlund [16] and Nair et al. [17], in two remarkable papers, proved independently Parisi's conjecture [18] about the average optimal cost at finite N . They were able to prove also the m...

We propose a class of mean-field models for the isostatic transition of systems of soft spheres, in which the contact network is modeled as a random graph and each contact is associated to d degrees of freedom. We study such models in the hypostatic, isostatic, and hyperstatic regimes. The density of states is evaluated by both the cavity method and exact diagonalization of the dynamical matrix. We show that the model correctly reproduces the main features of the density of states of real packings and, moreover, it predicts the presence of localized modes near the lower band edge. Finally, the behavior of the density of states D(ω)∼ω^{α} for ω→0 in the hyperstatic regime is studied. We find that the model predicts a nontrivial dependence of α on the details of the coordination distribution.

We consider the statistical inference problem of recovering an unknown perfect matching, hidden in a weighted random graph, by exploiting the information arising from the use of two different distributions for the weights on the edges inside and outside the planted matching. A recent work has demonstrated the existence of a phase transition, in the large size limit, between a full and a partial recovery phase for a specific form of the weights distribution on fully connected graphs. We generalize and extend this result in two directions: we obtain a criterion for the location of the phase transition for generic weights distributions and possibly sparse graphs, exploiting a technical connection with branching random walk processes, as well as a quantitatively more precise description of the critical regime around the phase transition.