Limit Theorem For Fluctuations Research Articles

On the Diffusive-Mean Field Limit for Weakly Interacting Diffusions Exhibiting Phase Transitions

The objective of this article is to analyse the statistical behaviour of a large number of weakly interacting diffusion processes evolving under the influence of a periodic interaction potential. We focus our attention on the combined mean field and diffusive (homogenisation) limits. In particular, we show that these two limits do not commute if the mean field system constrained to the torus undergoes a phase transition, that is to say, if it admits more than one steady state. A typical example of such a system on the torus is given by the noisy Kuramoto model of mean field plane rotators. As a by-product of our main results, we also analyse the energetic consequences of the central limit theorem for fluctuations around the mean field limit and derive optimal rates of convergence in relative entropy of the Gibbs measure to the (unique) limit of the mean field energy below the critical temperature.

On the Size of Chaos via Glauber Calculus in the Classical Mean-Field Dynamics

We consider a system of classical particles, interacting via a smooth, long-range potential, in the mean-field regime, and we optimally analyze the propagation of chaos in form of sharp estimates on many-particle correlation functions. While approaches based on the BBGKY hierarchy are doomed by uncontrolled losses of derivatives, we propose a novel non-hierarchical approach that focusses on the empirical measure of the system and exploits discrete stochastic calculus with respect to initial data in form of higher-order Poincare inequalities for cumulants. This main result allows to rigorously truncate the BBGKY hierarchy to an arbitrary precision on the mean-field timescale, thus justifying the Bogolyubov corrections to mean field. As corollaries, we also deduce a quantitative central limit theorem for fluctuations of the empirical measure, and we discuss the Lenard–Balescu limit for a spatially homogeneous system away from thermal equilibrium.

A Central limit theorem for fluctuations in Polyanalytic Ginibre ensembles

We study fluctuations of linear statistics in polyanalytic Ginibre ensembles, a family of point processes describing planar free fermions in a uniform magnetic field at higher Landau levels. Our ma ...

Functional central limit theorem for a class of negatively dependent heavy-tailed stationary infinitely divisible processes generated by conservative flows

We prove a functional central limit theorem for partial sums of symmetric stationary long-range dependent heavy tailed infinitely divisible processes. The limiting stable process is particularly interesting due to its long memory which is quantified by a Mittag–Leffler process induced by an associated Harris chain, at the discrete-time level. Previous results in Owada and Samorodnitsky [Ann. Probab. 43 (2015) 240–285] dealt with positive dependence in the increment process, whereas this paper derives the functional limit theorems under negative dependence. The negative dependence is due to cancellations arising from Gaussian-type fluctuations of functionals of the associated Harris chain. The new types of limiting processes involve stable random measures, due to heavy tails, Mittag–Leffler processes, due to long memory, and Brownian motions, due to the Gaussian second order cancellations. Along the way, we prove a function central limit theorem for fluctuations of functionals of Harris chains which is of independent interest as it extends a result of Chen [Probab. Theory Related Fields 116 (2000) 89–123].

Synchronization and fluctuation theorems for interacting Friedman urns

Abstract We consider a model of N interacting two-colour Friedman urns. The interaction model considered is such that the reinforcement of each urn depends on the fraction of balls of a particular colour in that urn as well as the overall fraction of balls of that colour in all the urns combined together. We show that the urns synchronize almost surely and that the fraction of balls of each colour converges to the deterministic limit of one-half, which matches with the limit known for a single Friedman urn. Furthermore, we use the notion of stable convergence to obtain limit theorems for fluctuations around the synchronization limit.

A central limit theorem for fluctuations in 1D stochastic homogenization

In this paper, we analyze the random fluctuations in a 1D stochastic homogenization problem and prove a central limit theorem: the first order fluctuations is described by a Gaussian process that solves an SPDE with an additive spatial white noise. Using a probabilistic approach, we obtain a precise error decomposition up to the first order, which also helps to decompose the limiting Gaussian process, with one of the components corresponding to the corrector obtained by a formal two-scale expansion.

Central limit theorem for fluctuations of linear eigenvalue statistics of large random graphs: Diluted regime

We study the linear eigenvalue statistics of large random graphs in the regimes when the mean number of edges for each vertex tends to infinity. We prove that for a rather wide class of test functions the fluctuations of linear eigenvalue statistics converges in distribution to a Gaussian random variable with zero mean and variance which coincides with “non-gaussian” part of the Wigner ensemble variance.

Central limit theorem for fluctuations of linear eigenvalue statistics of large random graphs

We consider the adjacency matrix A of a large random graph and study fluctuations of the function fn(z,u)=(1/n)∑k=1nexp{−uGkk(z)} with G(z)=(z−iA)−1. We prove that the moments of fluctuations normalized by n−1/2 in the limit n→∞ satisfy the Wick relations for the Gaussian random variables. This allows us to prove central limit theorem for Tr G(z) and then extend the result on the linear eigenvalue statistics Tr φ(A) of any function φ:R→R which increases, together with its first two derivatives, at infinity not faster than an exponential.

Some limit theorems for controlled branching processes

Limit theorems are obtained for branching processes depending on the size of the population. These results contain the case of convergence of such processes to the deterministic limit. Functional central limit theorems for fluctuations are proved.

Central limit theorem for random partitions under the Plancherel measure

In this work, we obtain the central limit theorem for fluctuations of Young diagrams around their limit shape in the bulk of the of partitions of a large integer n (under the Plancherel measure). More specifically, we show that, under the suitable normalization (growing as the square root of log n), the corresponding random process converges, in the sense of finite dimensional distributions, to a Gaussian process with independent values. The proof uses heavily the determinantal structure of the correlation functions and is based on the application of the Costin-Lebowitz-Soshnikov central limit theorem. At the spectrum edges, the fluctuation asymptotics is expressed in terms of the largest members of the Airy ensemble; in particular, at the upper edge the limit distribution appears to be discrete (without any normalization). These results admit an elegant symmetric reformulation under the rotation of Young diagrams by 45 degrees, where the normalization no longer depends on the location of the spectrum point. We also discuss the link of our central limit theorem with an earlier result by S.V. Kerov on the convergence to a generalized Gaussian process.

Gaussian fluctuations of Young diagrams under the Plancherel measure

We obtain the central limit theorem for fluctuations of Young diagrams around their limit shape in the bulk of the ‘spectrum’ of partitions λ ⊢ n ∈ (under the Plancherel measure), thus settling a long-standing problem posed by Logan & Shepp. Namely, under normalization growing like , the corresponding random process in the bulk is shown to converge, in the sense of finite-dimensional distributions, to a Gaussian process with independent values, while local correlations in the vicinity of each point, measured on various power scales, possess certain self-similarity. The proofs are based on the Poissonization techniques and use Costin–Lebowitz–Soshnikov's central limit theorem for determinantal random point processes. Our results admit a striking reformulation after the rotation of Young diagrams by 45°, whereby the normalization no longer depends on the location in the spectrum. In addition, we explain heuristically the link with an earlier result by Kerov on the convergence to a generalized Gaussian process.

Limit theorems for occupation time fluctuations of branching systems II: Critical and large dimensions

We give functional limit theorems for the fluctuations of the rescaled occupation time process of a critical branching particle system in R d with symmetric α -stable motion in the cases of critical and large dimensions, d = 2 α and d > 2 α . In a previous paper [T. Bojdecki, L.G. Gorostiza, A. Talarczyk, Limit theorems for occupation time fluctuations of branching systems I: long-range dependence, Stochastic Process. Appl., this issue.] we treated the case of intermediate dimensions, α < d < 2 α , which leads to a long-range dependence limit process. In contrast, in the present cases the limits are generalized Wiener processes. We use the same space–time random field method of the previous paper, the main difference being that now the tightness requires a new approach and the proofs are more difficult. We also give analogous results for the system without branching in the cases d = α and d > α .

Limit theorems for occupation time fluctuations of branching systems I: Long-range dependence

We give a functional limit theorem for the fluctuations of the rescaled occupation time process of a critical branching particle system in R d with symmetric α -stable motion and α < d < 2 α , which leads to a long-range dependence process involving sub-fractional Brownian motion. We also give an analogous result for the system without branching and d < α , which involves fractional Brownian motion. We use a space–time random field approach.

Central limit theorem for fluctuations in the high temperature region of the Sherrington–Kirkpatrick spin glass model

In a region above the Almeida–Thouless line, where we are able to control the thermodynamic limit of the Sherrington–Kirkpatrick model and to prove replica symmetry, we show that the fluctuations of the overlaps and of the free energy are Gaussian, on the scale 1/N, for large N. The method we employ is based on the idea we recently developed of introducing quadratic coupling between two replicas. The proof makes use of the cavity equations and of concentration of measure inequalities for the free energy.