Gabriele Sicuro

Scaling hypothesis for the Euclidean bipartite matching problem

We propose a simple yet very predictive form, based on a Poissons 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 monopartites one, is found for the subleading correction. We argue that the same scaling holds true for a generic cost exponent in dimension d > 2.

A theorem on the existence of symmetries of fractional PDEs

Abstract We propose a theorem that extends the classical Lie approach to the case of fractional partial differential equations (fPDEs) of the Riemann–Liouville type in ( 1 + 1 ) dimensions.

A foundational approach to the Lie theory for fractional order partial differential equations

Abstract We provide a general theoretical framework allowing us to extend the classical Lie theory for partial differential equations to the case of equations of fractional order. We propose a general prolongation formula for the study of Lie symmetries in the case of an arbitrary finite number of independent variables. We also prove the Lie theorem in the case of fractional differential equations, showing that the proper space for the analysis of these symmetries is the infinite dimensional jet space.

One-dimensional Euclidean matching problem: exact solutions, correlation functions, and universality.

We discuss the equivalence relation between the Euclidean bipartite matching problem on the line and on the circumference and the Brownian bridge process on the same domains. The equivalence allows us to compute the correlation function and the optimal cost of the original combinatorial problem in the thermodynamic limit; moreover, we solve also the minimax problem on the line and on the circumference. The properties of the average cost and correlation functions are discussed.

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

We analyze the random Euclidean bipartite matching problem on the hypertorus in d dimensions with quadratic cost and we derive the two-point correlation function for the optimal matching, using a proper ansatz introduced by Caracciolo et al. [Phys. Rev. E 90, 012118 (2014)] to evaluate the average optimal matching cost. We consider both the grid-Poisson matching problem and the Poisson-Poisson matching problem. We also show that the correlation function is strictly related to the Greens function of the Laplace operator on the hypertorus.

Quadratic Stochastic Euclidean Bipartite Matching Problem.

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 a particular case.

On the connection between linear combination of entropies and linear combination of extremizing distributions

Abstract We analyze the distribution that extremizes a linear combination of the Boltzmann–Gibbs entropy and the nonadditive q-entropy. We show that this distribution can be expressed in terms of a Lambert function. Both the entropic functional and the extremizing distribution can be associated with a nonlinear Fokker–Planck equation obtained from a master equation with nonlinear transition rates. Also, we evaluate the entropy extremized by a linear combination of a Gaussian distribution (which extremizes the Boltzmann–Gibbs entropy) and a q-Gaussian distribution (which extremizes the q-entropy). We give its explicit expression for q = 0 , and discuss the other cases numerically. The entropy that we obtain can be expressed, for q = 0 , in terms of Lambert functions, and exhibits a discontinuity in the second derivative for all values of q 1 . The entire discussion is closely related to recent results for type-II superconductors and for the statistics of the standard map.

Nonlinear inhomogeneous Fokker-Planck equations: Entropy and free-energy time evolution

We extend a recently introduced free-energy formalism for homogeneous Fokker-Planck equations to a wide, and physically appealing, class of inhomogeneous nonlinear Fokker-Planck equations. In our approach, the free-energy functional is expressed in terms of an entropic functional and an auxiliary potential, both derived from the coefficients of the equation. With reference to the introduced entropic functional, we discuss the entropy production in a relaxation process towards equilibrium. The properties of the stationary solutions of the considered Fokker-Planck equations are also discussed.

Finite-size corrections in the random assignment problem

We analytically derive, in the context of the replica formalism, the first finite-size corrections to the average optimal cost in the random assignment problem for a quite generic distribution law for the costs. We show that, when moving from a power-law distribution to a Γ distribution, the leading correction changes both in sign and in its scaling properties. We also examine the behavior of the corrections when approaching a δ-function distribution. By using a numerical solution of the saddle-point equations, we provide predictions that are confirmed by numerical simulations.

Free-energy formalism for inhomogeneous nonlinear Fokker-Planck equations

We extend the free-energy formalism recently introduced for homogeneous Fokker–Planck equations to a wide class of inhomogeneous nonlinear Fokker–Planck equations, providing sufficient conditions for the equation coefficients to obtain a free-energy that does not increase with time. Some properties of the stationary solutions of these Fokker–Planck equations are discussed.