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作者：Dereich, Steffen; Doring, Leif; Kyprianou, Andreas E.
作者单位：University of Munster; University of Mannheim; University of Bath
摘要：Since the seminal work of Lamperti, there is a lot of interest in the understanding of the general structure of self-similar Markov processes. Lamperti gave a representation of positive self-similar Markov processes with initial condition strictly larger than 0 which subsequently was extended to zero initial condition. For real self-sinfilar Markov processes (rssMps), there is a generalization of Lamperti's representation giving a one-to-one correspondence between Markov additive processes and...
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作者：Kemppainen, Antti; Smirnov, Stanislav
作者单位：University of Helsinki; University of Geneva
摘要：In this paper, we provide a framework of estimates for describing 2D scaling limits by Schramm's SLE curves. In particular, we show that a weak estimate on the probability of an annulus crossing implies that a random curve arising from a statistical mechanics model will have scaling limits and those will be well described by Loewner evolutions with random driving forces. Interestingly, our proofs indicate that existence of a nondegenerate observable with a conformally- invariant scaling limit ...
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作者：Kendall, Wilfrid S.
作者单位：University of Warwick
摘要：Consider an improper Poisson line process, marked by positive speeds so as to satisfy a scale-invariance property (actually, scale-equivariance). The line process can be characterized by its intensity measure, which belongs to a one-parameter family if scale and Euclidean invariance are required. This paper investigates a proposal by Aldous, namely that the line process could be used to produce a scale-invariant random spatial network (SIRSN) by means of connecting up points using paths which ...
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作者：Dembo, Amir; Tsai, Li-Cheng
作者单位：Stanford University; Stanford University; Columbia University
摘要：We study the fluctuation of the Atlas model, where a unit drift is assigned to the lowest ranked particle among a semi-infinite (Z(+)-indexed) system of otherwise independent Brownian particles, initiated according to a Poisson point process on R+. In this context, we show that the joint law of ranked particles, after being centered and scaled by t(-1/4), converges as t -> infinity to the Gaussian field corresponding to the solution of the Additive Stochastic Heat Equation (ASHE) on R+ with th...
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作者：Owada, Takashi; Adler, Robert J.
作者单位：Technion Israel Institute of Technology; Technion Israel Institute of Technology
摘要：We study the asymptotic nature of geometric structures formed from a point cloud of observations of (generally heavy tailed) distributions in a Euclidean space of dimension greater than one. A typical example is given by the Betti numbers of Cech complexes built over the cloud. The structure of dependence and sparcity (away from the origin) generated by these distributions leads to limit laws expressible via nonhomogeneous, random, Poisson measures. The parametrisation of the limits depends on...
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作者：Belinschi, Serban T.; Bercovici, Hari; Capitaine, Mireille; Fevrier, Maxime
作者单位：Centre National de la Recherche Scientifique (CNRS); Universite de Toulouse; Universite Toulouse III - Paul Sabatier; Queens University - Canada; Romanian Academy; University of Bucharest; Institute of Mathematics of the Romanian Academy; Indiana University System; Indiana University Bloomington; Universite Paris Saclay; Centre National de la Recherche Scientifique (CNRS)
摘要：We characterize the possible outliers in the spectrum of large deformed unitarily invariant additive and multiplicative models, as well as the eigenvectors corresponding to them. We allow both the nondeformed unitarily invariant model and the perturbation matrix to have nontrivial limiting eigenvalue distributions and spiked outliers in their spectrum. The free subordination functions play a key role in this analysis.
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作者：Benaych-Georges, Florent
作者单位：Universite Paris Cite
摘要：The single ring theorem, by Guionnet, Krishnapur and Zeitouni in Ann. of Math. (2) 174 (2011) 1189-1217, describes the empirical eigenvalue distribution of a large generic matrix with prescribed singular values, that is, an N x N matrix of the form A = UTV, with U, V some independent Haardistributed unitary matrices and T a deterministic matrix whose singular values are the ones prescribed. In this text, we give a local version of this result, proving that it remains true at the microscopic sc...
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作者：Ben-Hamou, Anna; Salez, Justin
作者单位：Universite Paris Cite; Universite Paris Cite
摘要：A finite ergodic Markov chain exhibits cutoff if its distance to stationarity remains close to 1 over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale. Discovered in the context of card shuffling (Aldous-Diaconis, 1986), this phenomenon is now believed to be rather typical among fast mixing Markov chains. Yet, establishing it rigorously often requires a challengingly detailed understanding of the underlying chain. Here, we consider nonbacktracking r...
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作者：Ren, Yan-Xia; Song, Renming; Zhang, Rui
作者单位：Peking University; Peking University; University of Illinois System; University of Illinois Urbana-Champaign
摘要：In this paper, we establish a spatial central limit theorem for a large class of supercritical branching, not necessarily symmetric, Markov processes with spatially dependent branching mechanisms satisfying a second moment condition. This central limit theorem generalizes and unifies all the central limit theorems obtained recently in Ren, Song and Zhang [J. Funct. Anal. 266 (2014) 1716-1756] for supercritical branching symmetric Markov processes. To prove our central limit theorem, we have to...
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作者：Hairer, Martin; Shen, Hao
作者单位：University of Warwick; University of Warwick; Columbia University
摘要：We consider the KPZ equation in one space dimension driven by a stationary centred space-time random field, which is sufficiently integrable and mixing, but not necessarily Gaussian. We show that, in the weakly asymmetric regime, the solution to this equation considered at a suitable large scale and in a suitable reference frame converges to the Hopf-Cole solution to the KPZ equation driven by space-time Gaussian white noise. While the limiting process depends only on the integrated variance o...
