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作者：Angel, Omer; Barlow, Martin T.; Gurel-Gurevich, Ori; Nachmias, Asaf
作者单位：University of British Columbia; Hebrew University of Jerusalem; Tel Aviv University
摘要：We provide a geometric representation of the Poisson and Martin boundaries of a transient, bounded degree triangulation of the plane in terms of its circle packing in the unit disc. (This packing is unique up to Mobius transformations.) More precisely, we show that any bounded harmonic function on the graph is the harmonic extension of some measurable function on the boundary of the disk, and that the space of extremal positive harmonic functions, that is, the Martin boundary, is homeomorphic ...
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作者：Lee, Ji Oon; Schnelli, Kevin; Stetler, Ben; Yau, Horng-Tzer
作者单位：Korea Advanced Institute of Science & Technology (KAIST); Institute of Science & Technology - Austria; Harvard University
摘要：We consider N x N random matrices of the form H = W + V where W is a real symmetric or complex Hermitian Wigner matrix and V is a random or deterministic, real, diagonal matrix whose entries are independent of W. We assume subexponential decay for the matrix entries of W, and we choose V so that the eigenvalues of W and V are typically of the same order. For a large class of diagonal matrices V, we show that the local statistics in the bulk of the spectrum are universal in the limit of large N.
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作者：Gouere, Jean-Baptiste; Marchand, Regine
作者单位：Universite de Orleans; Universite de Lorraine; Universite de Lorraine; Centre National de la Recherche Scientifique (CNRS)
摘要：Consider a Boolean model Sigma in R-d. The centers are given by a homogeneous Poisson point process with intensity lambda and the radii of distinct balls are i.i.d. with common distribution nu. The critical covered volume is the proportion of space covered by Sigma when the intensity lambda is critical for percolation. Previous numerical simulations and heuristic arguments suggest that the critical covered volume may be minimal when nu is a Dirac measure. In this paper, we prove that it is not...
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作者：Amir, Gideon; Angel, Omer; Benjamini, Itai; Kozma, Gady
作者单位：Bar Ilan University; University of British Columbia; Weizmann Institute of Science
摘要：We examine diffusion-limited aggregation generated by a random walk on Z with long jumps. We derive upper and lower bounds on the growth rate of the aggregate as a function of the number of moments a single step of the walk has. Under various regularity conditions on the tail of the step distribution, we prove that the diameter grows as n(beta+o(1)), with an explicitly given beta. The growth rate of the aggregate is shown to have three phase transitions, when the walk steps have finite third m...
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作者：Pimentel, Leandro P. R.
作者单位：Universidade Federal do Rio de Janeiro
摘要：In this article, we prove a duality relation between coalescence times and exit points in last-passage percolation models with exponential weights. As a consequence, we get lower bounds for coalescence times, with scaling exponent 3/2, and we relate its distribution with variational problems involving the Brownian motion process and the Airy(2) process. The proof relies on the relation between Busemann functions and the Burke property for stationary versions of the last-passage percolation mod...
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作者：Kahn, Jonas
作者单位：Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite de Lille
摘要：Aldous has introduced a notion of scale-invariant random spatial network (SIRSN) as a mathematical formalization of road networks. Intuitively, those are random processes that assign a route between each pair of points in Euclidean space, while being invariant under rotation, translation, and change of scale, and such that the routes are not too long and mainly lie on main roads. The only known example was somewhat artificial since invariance had to be added using randomization at the end of t...
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作者：Ekren, Ibrahim; Touzi, Nizar; Zhang, Jianfeng
作者单位：University of Southern California; Institut Polytechnique de Paris; Ecole Polytechnique
摘要：The main objective of this paper and the accompanying one [Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II (2012) Preprint] is to provide a notion of viscosity solutions for fully nonlinear parabolic path-dependent PDEs. Our definition extends our previous work [Ann. Probab. (2014) 42 204-236], focused on the semilinear case, and is crucially based on the nonlinear optimal stopping problem analyzed in [Stochastic Process. Appl. (2014) 124 3277-3311]. We prove that...
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作者：Buraczewski, Dariusz; Collamore, Jeffrey F.; Damek, Ewa; Zienkiewicz, Jacek
作者单位：University of Wroclaw; University of Copenhagen
摘要：In a variety of problems in pure and applied probability, it is relevant to study the large exceedance probabilities of the perpetuity sequence Y-n := B-1 + A(1)B(2) + ... + (A(1) ...A(n-1))B-n, where (A(i), B-i) subset of (0, infinity) x R. Estimates for the stationary tail distribution of {Y-n} have been developed in the seminal papers of Kesten [Acta Math. 131 (1973) 207-248] and Goldie [Ann. Appl. Probab. 1 (1991) 126-166]. Specifically, it is well known that if M := sup(n) Y-n, then P{M >...
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作者：Bloemendal, Alex; Virag, Balint
作者单位：Harvard University; University of Toronto
摘要：The top eigenvalues of rank r spiked real Wishart matrices and additively perturbed Gaussian orthogonal ensembles are known to exhibit a phase transition in the large size limit. We show that they have limiting distributions for near-critical perturbations, fully resolving the conjecture of Baik, Ben Arous and Peche [Duke Math. J. (2006) 133 205-235]. The starting point is a new (2r + 1)-diagonal form that is algebraically natural to the problem; for both models it converges to a certain rando...
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作者：Ekren, Ibrahim; Touzi, Nizar; Zhang, Jianfeng
作者单位：Swiss Federal Institutes of Technology Domain; ETH Zurich; Institut Polytechnique de Paris; Ecole Polytechnique; University of Southern California
摘要：In our previous paper [Ekren, Touzi and Zhang (2015)], we introduced a notion of viscosity solutions for fully nonlinear path-dependent PDEs, extending the semilinear case of Ekren et al. [Ann. Probab. 42 (2014) 204-236], which satisfies a partial comparison result under standard Lipshitz-type assumptions. The main result of this paper provides a full, well-posedness result under an additional assumption, formulated on some partial differential equation, defined locally by freezing the path. N...
