Probability Theory and Related Fields

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• The component graph of the uniform spanning forest: transitions in dimensions 9, 10, 11, ... AI翻译
  作者：Hutchcroft, Tom; Peres, Yuval
  作者单位：University of Cambridge; Microsoft
  摘要：We prove that the uniform spanning forests of Z(d) and Z(l) have qualitatively different connectivity properties whenever l > d >= 4. In particular, we consider the graph formed by contracting each tree of the uniform spanning forest down to a single vertex, which we call the component graph. We introduce the notion of ubiquitous subgraphs and show that the set of ubiquitous subgraphs of the component graph changes whenever the dimension changes and is above 8. To separate dimensions 5, 6, 7, ...
• Almost invariance of distributions for random walks on groups AI翻译
  作者：Erschler, Anna
  作者单位：Centre National de la Recherche Scientifique (CNRS); Universite PSL; Ecole Normale Superieure (ENS)
  摘要：We study the neighborhoods of a typical point Zn visited at n-th step of a random walk, determined by the condition that the transition probabilities stay close to n(Zn). If such neighborhood contains a ball of radius Cn, we say that the random walk has almost invariant transition probabilities. We prove that simple random walks on wreath products of Z with finite groups have almost invariant distributions. A weaker version of almost invariance implies a necessary condition of Ozawa's criterio...
• Rigidity of the three-dimensional hierarchical Coulomb gas AI翻译
  作者：Chatterjee, Sourav
  作者单位：Stanford University
  摘要：A random set of points in Euclidean space is called 'rigid' or 'hyperuniform' if the number of points falling inside any given region has significantly smaller fluctuations than the corresponding number for a set of i.i.d. random points. This phenomenon has received considerable attention in recent years, due to its appearance in random matrix theory, the theory of Coulomb gases and zeros of random analytic functions. However, most of the published results are in dimensions one and two. This p...
• The adaptive interpolation method: a simple scheme to prove replica formulas in Bayesian inference AI翻译
  作者：Barbier, Jean; Macris, Nicolas
  作者单位：Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne; Abdus Salam International Centre for Theoretical Physics (ICTP)
  摘要：In recent years important progress has been achieved towards proving the validity of the replica predictions for the (asymptotic) mutual information (or free energy) in Bayesian inference problems. The proof techniques that have emerged appear to be quite general, despite they have been worked out on a case-by-case basis. Unfortunately, a common point between all these schemes is their relatively high level of technicality. We present a new proof scheme that is quite straightforward with respe...
• Stationary solutions to the compressible Navier-Stokes system driven by stochastic forces AI翻译
  作者：Breit, Dominic; Feireisl, Eduard; Hofmanova, Martina; Maslowski, Bohdan
  作者单位：Heriot Watt University; Czech Academy of Sciences; Institute of Mathematics of the Czech Academy of Sciences; Technical University of Berlin; Charles University Prague
  摘要：We study the long-time behavior of solutions to a stochastically driven Navier-Stokes system describing the motion of a compressible viscous fluid driven by a temporal multiplicative white noise perturbation. The existence of stationary solutions is established in the framework of Lebesgue-Sobolev spaces pertinent to the class of weak martingale solutions. The methods are based on new global-in-time estimates and a combination of deterministic and stochastic compactness arguments. An essential...
• The Z-invariant Ising model via dimers AI翻译
  作者：Boutillier, Cedric; de Tiliere, Beatrice; Raschel, Kilian
  作者单位：Universite Paris Cite; Sorbonne Universite; Universite Paris-Est-Creteil-Val-de-Marne (UPEC); Centre National de la Recherche Scientifique (CNRS); Universite de Tours
  摘要：The Z-invariant Ising model(Baxter in Philos Trans R Soc Lond A Math Phys Eng Sci 289(1359):315-346, 1978) is defined on an isoradial graph and has coupling constants depending on an elliptic parameter k. When k=0 the model is critical, and as k varies the whole range of temperatures is covered. In this paper we study the corresponding dimer model on the Fisher graph, thus extending our papers(Boutillier and de Tiliere in Probab Theory Relat Fields 147:379-413, 2010; Commun Math Phys 301(2):47...
• Disorder and critical phenomena: the α=0 copolymer model AI翻译
  作者：Berger, Quentin; Giacomin, Giambattista; Lacoin, Hubert
  作者单位：Sorbonne Universite; Universite Paris Cite; Universite Paris Cite; Instituto Nacional de Matematica Pura e Aplicada (IMPA)
  摘要：The copolymer model is a disordered system built on a discrete renewal process with inter-arrival distribution that decays in a regularly varying fashion with exponent 1+alpha >= 1. It exhibits a localization transition which can be characterized in terms of the free energy of the model: the free energy is zero in the delocalized phase and it is positive in the localized phase. This transition, which is observed when tuning the mean h of the disorder variable, has been tackled in the physics l...
• Strong law of large numbers for the capacity of the Wiener sausage in dimension four AI翻译
  作者：Asselah, Amine; Schapira, Bruno; Sousi, Perla
  作者单位：Universite Paris-Est-Creteil-Val-de-Marne (UPEC); Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Aix-Marseille Universite; University of Cambridge
  摘要：We prove a strong law of large numbers for the Newtonian capacity of a Wiener sausage in the critical dimension four, where a logarithmic correction appears in the scaling. The main step of the proof is to obtain precise asymptotics for the expected value of the capacity. This requires a delicate analysis of intersection probabilities between two independent Wiener sausages.
• The law of large numbers for the maximum of almost Gaussian log-correlated fields coming from random matrices AI翻译
  作者：Lambert, Gaultier; Paquette, Elliot
  作者单位：University of Zurich; University System of Ohio; Ohio State University
  摘要：We compute the leading asymptotics as N of the maximum of the field QN(q)=log|q-AN|,qC, for any unitarily invariant Hermitian random matrix AN associated to a non-critical real-analytic potential. Hence, we verify the leading order in a conjecture ofFyodorov and Simm (Nonlinearity 29:2837, 2016. arXiv:1503.07110 [math-ph]) formulated for the GUE. The method relies on a classical upper-bound and a more sophisticated lower-bound based on a variant of the second-moment method which exploits the h...
• Polluted bootstrap percolation with threshold two in all dimensions AI翻译
  作者：Gravner, Janko; Holroyd, Alexander E.
  作者单位：University of California System; University of California Davis; University of Washington; University of Washington Seattle
  摘要：In the polluted bootstrap percolation model, the vertices of a graph are independently declared initially occupied with probability p or closed with probability q. At subsequent steps, a vertex becomes occupied if it is not closed and it has at least r occupied neighbors. On the cubic lattice Z(d) of dimension d >= 3 with threshold r = 2, we prove that the final density of occupied sites converges to 1 as p and q both approach 0, regardless of their relative scaling. Our result partially resol...