Probability Theory and Related Fields最新文献

We study the scaling properties of the non-equilibrium stationary states (NESS) of a reaction-diffusion model. Under a suitable smallness condition, we show that the density of particles satisfies a law of large numbers with respect to the NESS, with an explicit rate of convergence, and we also show that at mesoscopic scales the NESS is well approximated by a local equilibrium (product) measure, in the total variation distance. In addition, in dimensions (d le 3) we show a central limit theorem for the density of particles under the NESS. The corresponding Gaussian limit can be represented as an independent sum of a white noise and a massive Gaussian free field, and in particular it presents macroscopic correlations.

Consider an infinite collection of particles on the real line moving according to independent Brownian motions and such that the i-th particle from the left gets the drift (g_{i-1}). The case where (g_0=1) and (g_{i}=0) for all (i in {mathbb {N}}) corresponds to the well studied infinite Atlas model. Under conditions on the drift vector ({varvec{g}}= (g_0, g_1, ldots )') it is known that the Markov process corresponding to the gap sequence of the associated ranked particles has a continuum of product form stationary distributions ({pi _a^{{varvec{g}}}, a in S^{{varvec{g}}}}) where (S^{{varvec{g}}}) is a semi-infinite interval of the real line. In this work we show that all of these stationary distributions are extremal and ergodic. We also prove that any product form stationary distribution of this Markov process that satisfies a mild integrability condition must be (pi _a^{{varvec{g}}}) for some (a in S^{{varvec{g}}}). These results are new even for the infinite Atlas model. The work makes progress on the open problem of characterizing all the invariant distributions of general competing Brownian particle systems interacting through their relative ranks. Proofs rely on synchronous and mirror coupling of Brownian particles and properties of the intersection local times of the various particles in the infinite system.

In this paper, we study the smallest non-zero eigenvalue of the sample covariance matrices (mathcal {S}(Y)=YY^*), where (Y=(y_{ij})) is an (Mtimes N) matrix with iid mean 0 variance (N^{-1}) entries. We consider the regime (M=M(N)) and (M/Nrightarrow c_infty in mathbb {R}{setminus } {1}) as (Nrightarrow infty ). It is known that for the extreme eigenvalues of Wigner matrices and the largest eigenvalue of (mathcal {S}(Y)), a weak 4th moment condition is necessary and sufficient for the Tracy–Widom law (Ding and Yang in Ann Appl Probab 28(3):1679–1738, 2018. https://doi.org/10.1214/17-AAP1341; Lee and Yin in Duke Math J 163(1):117–173, 2014. https://doi.org/10.1215/00127094-2414767). In this paper, we show that the Tracy–Widom law is more robust for the smallest eigenvalue of (mathcal {S}(Y)), by discovering a phase transition induced by the fatness of the tail of (y_{ij})’s. More specifically, we assume that (y_{ij}) is symmetrically distributed with tail probability (mathbb {P}(|sqrt{N}y_{ij}|ge x)sim x^{-alpha }) when (xrightarrow infty ), for some (alpha in (2,4)). We show the following conclusions: (1) When (alpha >frac{8}{3}), the smallest eigenvalue follows the Tracy–Widom law on scale (N^{-frac{2}{3}}); (2) When (2<alpha <frac{8}{3}), the smallest eigenvalue follows the Gaussian law on scale (N^{-frac{alpha }{4}}); (3) When (alpha =frac{8}{3}), the distribution is given by an interpolation between Tracy–Widom and Gaussian; (4) In case (alpha le frac{10}{3}), in addition to the left edge of the MP law, a deterministic shift of order (N^{1-frac{alpha }{2}}) shall be subtracted from the smallest eigenvalue, in both the Tracy–Widom law and the Gaussian law. Overall speaking, our proof strategy is inspired by Aggarwal et al. (J Eur Math Soc 23(11):3707–3800, 2021. https://doi.org/10.4171/jems/1089) which is originally done for the bulk regime of the Lévy Wigner matrices. In addition to various technical complications arising from the bulk-to-edge extension, two ingredients are needed for our derivation: an intermediate left edge local law based on a simple but effective matrix minor argument, and a mesoscopic CLT for the linear spectral statistic with asymptotic expansion for its expectation.

Given finite i.i.d. samples in a Hilbert space with zero mean and trace-class covariance operator (Sigma ), the problem of recovering the spectral projectors of (Sigma ) naturally arises in many applications. In this paper, we consider the problem of finding distributional approximations of the spectral projectors of the empirical covariance operator ({hat{Sigma }}), and offer a dimension-free framework where the complexity is characterized by the so-called relative rank of (Sigma ). In this setting, novel quantitative limit theorems and bootstrap approximations are presented subject to mild conditions in terms of moments and spectral decay. In many cases, these even improve upon existing results in a Gaussian setting.

20 years ago, Bovier, Kurkova, and Löwe (Ann Probab 30(2):605–651, 2002) proved a central limit theorem (CLT) for the fluctuations of the free energy in the p-spin version of the Sherrington–Kirkpatrick model of spin glasses at high temperatures. In this paper we improve their results in two ways. First, we extend the range of temperatures to cover the entire regime where the quenched and annealed free energies are known to coincide. Second, we identify the main source of the fluctuations as a purely coupling dependent term, and we show a further CLT for the deviation of the free energy around this random object.

For the KPZ equation on a torus with a (1+1) spacetime white noise, it was shown in Dunlap et al. (Commun Pure Appl Math, 2023, https://doi.org/10.1002/cpa.22110) and Gu and Komorowski (Ann Inst H Poincare Prob Stat, 2021, arXiv:2104.13540v2) that the height function satisfies a central limit theorem, and the variance can be written as the expectation of an exponential functional of Brownian bridges. In this paper, we consider another physically relevant quantity, the winding number of the directed polymer on a cylinder, or equivalently, the displacement of the directed polymer endpoint in a spatially periodic random environment. It was shown in Gu and Komorowski (SIAM J Math Anal, arXiv:2207.14091) that the polymer endpoint satisfies a central limit theorem on diffusive scales. The main result of this paper is an explicit expression of the effective diffusivity, in terms of the expectation of another exponential functional of Brownian bridges. Our argument is based on a combination of tools from Malliavin calculus, homogenization, and diffusion in distribution-valued random environments.

We prove a central limit theorem (CLT) for the Fréchet mean of independent and identically distributed observations in a compact Riemannian manifold assuming that the population Fréchet mean is unique. Previous general CLT results in this setting have assumed that the cut locus of the Fréchet mean lies outside the support of the population distribution. In this paper we present a CLT under some mild technical conditions on the manifold plus the following assumption on the population distribution: in a neighbourhood of the cut locus of the population Fréchet mean, the population distribution is absolutely continuous with respect to the volume measure on the manifold and in this neighhbourhood the Radon–Nikodym derivative has a version that is continuous. So far as we are aware, the CLT given here is the first which allows the cut locus to have co-dimension one or two when it is included in the support of the distribution. A key part of the proof is establishing an asymptotic approximation for the parallel transport of a certain vector field. Whether or not a non-standard term arises in the CLT depends on whether the co-dimension of the cut locus is one or greater than one: in the former case a non-standard term appears but not in the latter case. This is the first paper to give a general and explicit expression for the non-standard term which arises when the co-dimension of the cut locus is one.

We consider the nonlinear Schrödinger equation with multiplicative spatial white noise and an arbitrary polynomial nonlinearity on the two-dimensional full space domain. We prove global well-posedness by using a gauge-transform introduced by Hairer and Labbé (Electron Commun Probab 20(43):11, 2015) and constructing the solution as a limit of solutions to a family of approximating equations. This paper extends a previous result by Debussche and Martin (Nonlinearity 32(4):1147–1174, 2019) with a sub-quadratic nonlinearity.

For a large class of amenable transient weighted graphs G, we prove that the sign clusters of the Gaussian free field on G fall into a regime of strong supercriticality, in which two infinite sign clusters dominate (one for each sign), and finite sign clusters are necessarily tiny, with overwhelming probability. Examples of graphs belonging to this class include regular lattices such as ({mathbb {Z}}^d), for (dge 3), but also more intricate geometries, such as Cayley graphs of suitably growing (finitely generated) non-Abelian groups, and cases in which random walks exhibit anomalous diffusive behavior, for instance various fractal graphs. As a consequence, we also show that the vacant set of random interlacements on these objects, introduced by Sznitman (Ann Math 171(3):2039–2087, 2010), and which is intimately linked to the free field, contains an infinite connected component at small intensities. In particular, this result settles an open problem from Sznitman (Invent Math 187(3):645–706, 2012).

We give a novel characterization of the centered model in regularity structures which persists for rough drivers even as a mollification fades away. We present our result for a class of quasilinear equations driven by noise, however we believe that the method is robust and applies to a much broader class of subcritical equations. Furthermore, we prove that a convergent sequence of noise ensembles, satisfying uniformly a spectral gap assumption, implies the corresponding convergence of the associated models. Combined with the characterization, this establishes a universality-type result.