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Gabriele Sicuro

Publications and source records attributed to Gabriele Sicuro.

32 records · Page 2Linked to original sources
arXiv2017

Random Euclidean matching problems in one dimension

We discuss the optimal matching solution for both the assignment problem and the matching problem in one dimension for a large class of convex cost functions. We consider the problem in a compact set with the topology both of the interval and of the circumference. Afterwards, we assume the points' positions to be random variables identically and independently distributed on the considered domain. We analytically obtain the average optimal cost in the asymptotic regime of very large number of points $N$ and some correlation functions for a power-law type cost function in the form $c(z)=z^p$, both in the $p>1$ case and in the $p<0$ case. The scaling of the optimal mean cost with the number of points is $N^{-\frac{p}{2}}$ for the assignment and $N^{-p}$ for the matching when $p>1$, whereas in both cases it is a constant when $p<0$. Finally, our predictions are compared with the results of numerical simulations.

cond-mat.dis-nn↗
arXiv2017

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.

cond-mat.dis-nn↗
arXiv2017

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

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.

math.AP↗
arXiv2016

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.

cond-mat.stat-mech↗
arXiv2016

One-loop diagrams in the Random Euclidean Matching Problem

The matching problem is a notorious combinatorial optimization problem that has attracted for many years the attention of the statistical physics community. Here we analyze the Euclidean version of the problem, i.e. the optimal matching problem between points randomly distributed on a $d$-dimensional Euclidean space, where the cost to minimize depends on the points' pairwise distances. Using Mayer's cluster expansion we write a formal expression for the replicated action that is suitable for a saddle point computation. We give the diagrammatic rules for each term of the expansion, and we analyze in detail the one-loop diagrams. A characteristic feature of the theory, when diagrams are perturbatively computed around the mean field part of the action, is the vanishing of the mass at zero momentum. In the non-Euclidean case of uncorrelated costs instead, we predict and numerically verify an anomalous scaling for the sub-sub-leading correction to the asymptotic average cost.

cond-mat.dis-nn↗
arXiv2016

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

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.

cond-mat.stat-mech↗
arXiv2016

Groups, Information Theory and Einstein's Likelihood Principle

We propose a unifying picture where the notion of generalized entropy is related to information theory by means of a group-theoretical approach. The group structure comes from the requirement that an entropy be well defined with respect to the composition of independent systems, in the context of a recently proposed generalization of the Shannon-Khinchin axioms. We associate to each member of a large class of entropies a generalized information measure, satisfying the additivity property on a set of independent systems as a consequence of the underlying group law. At the same time, we also show that Einstein's likelihood function naturally emerges as a byproduct of our informational interpretation of (generally nonadditive) entropies. These results confirm the adequacy of composable entropies both in physical and social science contexts.

cond-mat.stat-mech↗
arXiv2015

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, $N\gg 1$. The points are supposed independently randomly generated on a domain $Ω\subset\mathbb R^d$ with a given distribution $ρ(\mathbf x)$ 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.

cond-mat.stat-mech↗
arXiv2015

On the robustness of the $q$-Gaussian family

We introduce three deformations, called $α$-, $β$- and $γ$-deformation respectively, of a $N$-body probabilistic model, first proposed by Rodríguez et al. (2008), having $q$-Gaussians as $N\to\infty$ limiting probability distributions. The proposed $α$- and $β$-deformations are asymptotically scale-invariant, whereas the $γ$-deformation is not. We prove that, for both $α$- and $β$-deformations, the resulting deformed triangles still have $q$-Gaussians as limiting distributions, with a value of $q$ independent (dependent) on the deformation parameter in the $α$-case ($β$-case). In contrast, the $γ$-case, where we have used the celebrated $Q$-numbers and the Gauss binomial coefficients, yields other limiting probability distribution functions, outside the $q$-Gaussian family. These results suggest that scale-invariance might play an important role regarding the robustness of the $q$-Gaussian family.

cond-mat.stat-mech↗
arXiv2015

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. 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 Green's function of the Laplace operator on the hypertorus.

cond-mat.dis-nn↗
arXiv2014

Scaling hypothesis for the Euclidean bipartite matching problem

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 non-trivial scaling exponent, $γ_d=\frac{d-2}{d}$, 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 $d>2$.

cond-mat.dis-nn↗
arXiv2014

On the 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 combinatoric 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.

cond-mat.dis-nn↗