A large deviation principle for empirical measures on Polish spaces: Application to singular Gibbs measures on manifolds

García-Zelada, David

doi:10.1214/18-aihp922

Cited by 31 publications

(28 citation statements)

References 23 publications

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“…Moreover, quite a few equilibrium measures are known for log-gases beyond Coulomb gases, see for instance [14]. Actually it can be shown that essentially if β N N and V beats g at infinity then under (P N ) N the sequence of random empirical measures (µ N ) N satisfies a large deviation principle with speed β N and good rate function E, see [12,30,3]. Concentration of measure inequalities are also available, see [15] and references therein.…”

Section: Static Energy and Equilibrium Measuresmentioning

confidence: 99%

Simulating Coulomb and Log-Gases with Hybrid Monte Carlo Algorithms

Chafäı

Ferré

2018

J Stat Phys

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Coulomb and log-gases are exchangeable singular Boltzmann-Gibbs measures appearing in mathematical physics at many places, in particular in random matrix theory. We explore experimentally an efficient numerical method for simulating such gases. It is an instance of the Hybrid or Hamiltonian Monte Carlo algorithm, in other words a Metropolis-Hastings algorithm with proposals produced by a kinetic or underdamped Langevin dynamics. This algorithm has excellent numerical behavior despite the singular interaction, in particular when the number of particles gets large. It is more efficient than the well known overdamped version previously used for such problems, and allows new numerical explorations. It suggests for instance to conjecture a universality of the Gumbel fluctuation at the edge of beta Ginibre ensembles for all beta.

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Section: Static Energy and Equilibrium Measuresmentioning

confidence: 99%

Simulating Coulomb and Log-Gases with Hybrid Monte Carlo Algorithms

Chafäı

Ferré

2018

J Stat Phys

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“…For pairwise interactions, i.e., F (ν) = ν ⊗2 , V for some V : E 2 → R, this large deviation principle is now known to hold under much broader assumptions which cover singular interactions V ; these results require much more care, particularly when the temperature is allowed to depend on n [11,19,35,22]. When F + H(• | λ) admits a unique minimizer µ, the large deviation principle implies the law of large numbers…”

Section: Mean Field Gibbs Measuresmentioning

confidence: 99%

Quantitative approximate independence for continuous mean field Gibbs measures

Lacker

2022

Electron. J. Probab.

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Many Gibbs measures with mean field interactions are known to be chaotic, in the sense that any collection of k particles in the n-particle system are asymptotically independent, as n → ∞ with k fixed or perhaps k = o(n). This paper quantifies this notion for a class of continuous Gibbs measures on Euclidean space with pairwise interactions, with main examples being systems governed by convex interactions and uniformly convex confinement potentials. The distance between the marginal law of k particles and its limiting product measure is shown to be O((k/n) c∧2 ), with c proportional to the squared temperature. In the high temperature case, this improves upon prior results based on subadditivity of entropy, which yield O(k/n) at best. The bound O((k/n) 2 ) cannot be improved, as a Gaussian example demonstrates. The results are non-asymptotic, and distance is quantified via relative Fisher information, relative entropy, or the squared quadratic Wasserstein metric. The method relies on an a priori functional inequality for the limiting measure, used to derive an estimate for the k-particle distance in terms of the (k + 1)-particle distance.

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“…a lower bound on the partition function K n,Λ,V itself. In some cases one can get a lower bound on the partition function almost "for free" using Jensen's inequality (see [Gar19a]) but that trick is not suitable here because of the lack of control of V in L ∞ (Λ). We resort to an "explicit" construction.…”

Section: Finishing the Proofmentioning

confidence: 99%

The two-dimensional one-component plasma is hyperuniform

Leblé¹

2021

Preprint

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We prove that at all positive temperatures in the bulk of a classical two-dimensional one-component plasma (also called Coulomb or log-gas, or jellium) the variance of the number of particles in large disks grows more slowly than the area. In other words the system is hyperuniform.We obtain a non-sharp but quantitative bound on the number variance's growth rate, which is the first mathematical justification of an old prediction in the physics literature about "suppression of charge fluctuations".We introduce an argument of approximate conditional independence for well-separated sub-systems and a trick using "isotropically averaged localized translations" in order to control the expectation of non-smooth linear statistics.1 Our argument works for α > 31 32 .

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A large deviation principle for empirical measures on Polish spaces: Application to singular Gibbs measures on manifolds

Cited by 31 publications

References 23 publications

Simulating Coulomb and Log-Gases with Hybrid Monte Carlo Algorithms

Simulating Coulomb and Log-Gases with Hybrid Monte Carlo Algorithms

Quantitative approximate independence for continuous mean field Gibbs measures

The two-dimensional one-component plasma is hyperuniform

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