Probability Theory and Related Fields最新文献

We propose a new concept of lifts of reversible diffusion processes and show that various well-known non-reversible Markov processes arising in applications are lifts in this sense of simple reversible diffusions. Furthermore, we introduce a concept of non-asymptotic relaxation times and show that these can at most be reduced by a square root through lifting, generalising a related result in discrete time. Finally, we demonstrate how the recently developed approach to quantitative hypocoercivity based on space–time Poincaré inequalities can be rephrased and simplified in the language of lifts and how it can be applied to find optimal lifts.

We consider coupled slow-fast stochastic processes, where the averaged slow motion is given by a two-dimensional Hamiltonian system with multiple critical points. On a proper time scale, the evolution of the first integral converges to a diffusion process on the corresponding Reeb graph, with certain gluing conditions specified at the interior vertices, as in the case of additive white noise perturbations of Hamiltonian systems considered by M. Freidlin and A. Wentzell. The current paper provides the first result where the motion on a graph and the corresponding gluing conditions appear due to the averaging of a slow-fast system, with a Hamiltonian structure, on a large time scale. The result allows one to consider, for instance, long-time diffusion approximation for an oscillator with a potential with more than one well.

In this paper, we study the critical level-set of Gaussian free field (GFF) on the metric graph (widetilde{{mathbb {Z}}}^d,d>6). We prove that the one-arm probability (i.e. the probability of the event that the origin is connected to the boundary of the box B(N)) is proportional to (N^{-2}), where B(N) is centered at the origin and has side length (2lfloor N rfloor ). Our proof is highly inspired by Kozma and Nachmias (J Am Math Soc 24(2):375–409, 2011) which proves the analogous result for the critical bond percolation for (dge 11), and by Werner (in: Séminaire de Probabilités XLVIII, Springer, Berlin, 2016) which conjectures the similarity between the GFF level-set and the bond percolation in general and proves this connection for various geometric aspects.

This paper considers the effect of additive white noise on the normal form for the supercritical Hopf bifurcation in 2 dimensions. The main results involve the asymptotic behavior of the top Lyapunov exponent (lambda ) associated with this random dynamical system as one or more of the parameters in the system tend to 0 or (infty ). This enables the construction of a bifurcation diagram in parameter space showing stable regions where (lambda <0) (implying synchronization) and unstable regions where (lambda > 0) (implying chaotic behavior). The value of (lambda ) depends strongly on the shearing effect of the twist factor b/a of the deterministic Hopf bifurcation. If b/a is sufficiently small then (lambda <0) regardless of all the other parameters in the system. But when all the parameters except b are fixed then (lambda ) grows like a positive multiple of (b^{2/3}) as (b rightarrow infty ).

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.