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Emanuele Caglioti

Publications and source records attributed to Emanuele Caglioti.

At least 19 recordsLinked to original sources
arXiv2026

Time Averages for the Vortex Model and Stroboscopic Ergodic Averages

We consider the vortex model on the plane, focusing on the case of vortices with the same sign and, for simplicity, assuming all vortices possess equal circulation. In particular we are interested at the time average of the vorticity density, i.e. the empirical measure associated to the vortices. We conjecture that, for a.e. initial data, the time average of the empirical density is radial. We prove the result for N=3 vortices by exploiting the integrability of the system. For N > 3 vortices we motivate the conjecture by transforming the problem into the independence of ergodic stroboscopic averages from initial data along a single trajectory, when using a suitable rotation angle as the independent variable instead of the time variable.

math-ph↗
arXiv2024

Subadditivity and optimal matching of unbounded samples

We obtain new bounds for the optimal matching cost for empirical measures with unbounded support. For a large class of radially symmetric and rapidly decaying probability laws, we prove for the first time the asymptotic rate of convergence for the whole range of power exponents $p$ and dimensions $d$. Moreover we identify the exact prefactor when $p\le d$. We cover in particular the Gaussian case, going far beyond the currently known bounds. Our proof technique is based on approximate sub- and super-additivity bounds along a geometric decomposition adapted to some features the density, such as its radial symmetry and its decay at infinity.

math.PR↗
arXiv2024

Scattering problem for Vlasov-type equations on the $d$-dimensional torus with Gevrey data

In this article, we consider Vlasov-type equations describing the evolution of single-species type plasmas, such as those composed of electrons (Vlasov-Poisson) or ions (screened Vlasov-Poisson/Vlasov-Poisson with massless electrons). We solve the final data problem on the torus $\mathbb{T}^d$, $d \geq 1$, by considering asymptotic states of regularity Gevrey-$\frac{1}γ$ with $γ>\frac13$, small perturbations of homogeneous equilibria satisfying the Penrose stability condition. This extends to the Gevrey perturbative case, and to higher dimension, the scattering result in analytic regularity obtained by E. Caglioti and C. Maffei in [14], and answers an open question raised by J. Bedrossian in arXiv:2211.13707.

math.AP↗
arXiv2024

Random matching in 2D with exponent 2 for gaussian densities

We solve the Random Euclidean Matching problem with exponent 2 for the Gaussian distribution defined on the plane. Previous works by Ledoux and Talagrand determined the leading behavior of the average cost up to a multiplicative constant. We explicitly determine the constant, showing that the average cost is proportional to (log N)^2, where N is the number of points. Our approach relies on a geometric decomposition allowing an explicit computation of the constant. Our results illustrate the potential for exact solutions of random matching problems for many distributions defined on unbounded domains on the plane.

math.PR↗
arXiv2023

Microcanonical phase transitions for the vortex system

We consider the Microcanonical Variational Principle for the vortex system in a bounded domain. In particular we are interested in the thermodynamic properties of the system in domains of second kind, i.e. for which the equivalence of ensembles does not hold. For connected domains close to the union of disconnected disks (dumbbell domains), we show that the system may exhibit an arbitrary number of fist-order phase transitions, while the entropy is convex for large energy.

math-ph↗
arXiv2022

Asymptotic probability of energy increasing solutions to the homogeneous Boltzmann equation

Weak solutions to the homogeneous Boltzmann equation with increasing energy have been constructed by Lu and Wennberg. We consider an underlying microscopic stochastic model with binary collision (Kac's model) and show that these solutions are atypical. More precisely, we prove that the probability of observing these paths is exponentially small in the number of particles and compute the exponential rate. This result is obtained by improving the established large deviation estimates in the canonical setting. Key ingredients are the extension of Sanov's theorem to the microcanonical ensemble and large deviations for the Kac's model in the microcanonical setting.

math-ph↗
arXiv2021

Comparison between the Cauchy problem and the scattering problem for the Landau damping in the Vlasov-HMF equation

We analyze the analytic Landau damping problem for the Vlasov-HMF equation, by fixing the asymptotic behavior of the solution. We use a new method for this "scattering problem", closer to the one used for the Cauchy problem. In this way we are able to compare the two results, emphasizing the different influence of the plasma echoes in the two approaches. In particular, we prove a non-perturbative result for the scattering problem.

math.AP↗
arXiv2021

Mean-field limit for particle systems with topological interactions

The mean-field limit for systems of self-propelled agents with topological interaction cannot be obtained by means of the usual Dobrushin approach. We get a result on this direction by adapting to the multidimensional case the techniques developed by Trocheris in 1986 to treat the Vlasov-Poisson equation in one dimension.

math.AP↗
arXiv2021

Random assignment problems on ${2d}$ manifolds

We consider the assignment problem between two sets of $N$ random points on a smooth, two-dimensional manifold $Ω$ of unit area. It is known that the average cost scales as $E_Ω(N)\sim\frac{1}{2π}\ln N$ with a correction that is at most of order $\sqrt{\ln N\ln\ln N}$. In this paper, we show that, within the linearization approximation of the field-theoretical formulation of the problem, the first $Ω$-dependent correction is on the constant term, and can be exactly computed from the spectrum of the Laplace--Beltrami operator on $Ω$. We perform the explicit calculation of this constant for various families of surfaces, and compare our predictions with extensive numerics.

math-ph↗
arXiv2021

Quantum optimal transport is cheaper

We compare bipartite (Euclidean) matching problems in classical and quantum mechanics. The quantum case is treated in terms of a quantum version of the Wasserstein distance introduced in [F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. 343 (2016), 165-205]. We show that the optimal quantum cost can be cheaper than the classical one. We treat in detail the case of two particles: the equal mass case leads to equal quantum and classical costs. Moreover, we show examples with different masses for which the quantum cost is strictly cheaper than the classical cost.

math.AP↗
arXiv2021

Towards Optimal Transport for Quantum Densities

An analogue of the quadratic Wasserstein (or Monge-Kantorovich) distance between Borel probability measures on $\mathbf{R}^d$ has been defined in [F. Golse, C. Mouhot, T. Paul: Commun. Math. Phys. 343 (2015), 165-205] for density operators on $L^2(\mathbf{R}^d)$, and used to estimate the convergence rate of various asymptotic theories in the context of quantum mechanics. The present work proves a Kantorovich type duality theorem for this quantum variant of the Monge-Kantorovich or Wasserstein distance, and discusses the structure of optimal quantum couplings. Specifically, we prove that optimal quantum couplings involve a gradient type structure similar to the Brenier transport map (which is the gradient of a convex function), or more generally, to the subdifferential of a l.s.c. convex function as in the Knott-Smith optimality criterion (see Theorem 2.12 in [C. Villani: "Topics in Optimal Transportation", Amer. Math. Soc. 2003]).

math-ph↗
arXiv2020

Euclidean random matching in 2D for non-constant densities

We consider the 2-dimensional random matching problem in $\mathbb{R}^2.$ In a challenging paper, Caracciolo et. al. arXiv:1402.6993 on the basis of a subtle linearization of the Monge Ampere equation, conjectured that the expected value of the square of the Wasserstein distance, with exponent $2,$ between two samples of $N$ uniformly distributed points in the unit square is $\log N/2πN$ plus corrections, while the expected value of the square of the Wasserstein distance between one sample of $N$ uniformly distributed points and the uniform measure on the square is $\log N/4πN$. These conjectures has been proved by Ambrosio et al. arXiv:1611.04960. Here we consider the case in which the points are sampled from a non uniform density. For first we give formal arguments leading to the conjecture that if the density is regular and positive in a regular, bounded and connected domain $Λ$ in the plane, then the leading term of the expected values of the Wasserstein distances are exactly the same as in the case of uniform density, but for the multiplicative factor equal to the measure of $Λ$. We do not prove these results but, in the case in which the domain is a square, we prove estimates from above that coincides with the conjectured result.

math-ph↗
arXiv2019

Some aspects of the inertial spin model for flocks and related kinetic equations

In this paper we study the macroscopic behavior of the inertial spin (IS) model. This model has been recently proposed to describe the collective dynamics of flocks of birds, and its main feature is the presence of an auxiliary dynamical variable, a sort of internal spin, which conveys the interaction among the birds with the effect of better describing the turning of flocks. After discussing the geometrical and mechanical properties of the IS model, we show that, in the case of constant interaction among the birds, its mean-field limit is described by a non-linear Fokker-Planck equation, whose equilibria are fully characterized. Finally, in the case of non-constant interactions, we derive the kinetic equation for the mean-field limit of the model in the absence of thermal noise, and explore its macroscopic behavior by analyzing the mono-kinetic solutions.

math-ph↗
arXiv2016

Quantization of Measures and Gradient Flows: a Perturbative Approach in the 2-Dimensional Case

In this paper we study a perturbative approach to the problem of quantization of measures in the plane. Motivated by the fact that, as the number of points tends to infinity, hexagonal lattices are asymptotically optimal from an energetic point of view (see [Morgan, Bolton: Amer. Math. Monthly 109 (2002), 165-172]), we consider configurations that are small perturbations of the hexagonal lattice and we show that: (1) in the limit as the number of points tends to infinity, the hexagonal lattice is a strict minimizer of the energy; (2) the gradient flow of the limiting functional allows us to evolve any perturbed configuration to the optimal one exponentially fast. In particular, our analysis provides a solid mathematical justification of the asymptotic optimality of the hexagonal lattice among its nearby configurations.

math.AP↗
arXiv2016

Translation invariant realizability problem on the $d-$dimensional lattice: an explicit construction

We consider a particular instance of the truncated realizability problem on the $d-$dimensional lattice. Namely, given two functions $ρ_1({\bf i})$ and $ρ_2({\bf i},{\bf j})$ non-negative and symmetric on $\mathbb{Z}^d$, we ask whether they are the first two correlation functions of a translation invariant point process. We provide an explicit construction of such a realizing process for any $d\geq 2$ when the radial distribution has a specific form. We also derive from this construction a lower bound for the maximal realizable density and compare it with the already known lower bounds.

math.PR↗
arXiv2015

A gradient flow approach to quantization of measures

In this paper we study a gradient flow approach to the problem of quantization of measures in one dimension. By embedding our problem in $L^2$, we find a continuous version of it that corresponds to the limit as the number of particles tends to infinity. Under some suitable regularity assumptions on the density, we prove uniform stability and quantitative convergence result for the discrete and continuous dynamics.

math.AP↗
arXiv2014

Exponential dephasing of oscillators in the Kinetic Kuramoto Model

We study the kinetic Kuramoto model for coupled oscillators with coupling constant below the synchronization threshold. We manage to prove that, for any analytic initial datum, if the interaction is small enough, the order parameter of the model vanishes exponentially fast, and the solution is asymptotically described by a free flow. This behavior is similar to the phenomenon of Landau damping in plasma physics. In the proof we use a combination of techniques from Landau damping and from abstract Cauchy-Kowalewskaya theorem.

math-ph↗