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作者：Ben Arous, Gerard; Bourgade, Paul
作者单位：New York University; Harvard University
摘要：This paper studies the extreme gaps between eigenvalues of random matrices. We give the joint limiting law of the smallest gaps for Haar-distributed unitary matrices and matrices from the Gaussian unitary ensemble. In particular, the kth smallest gap, normalized by a factor n(-4/3), has a limiting density proportional to x(3k-1)e(-x3). Concerning the largest gaps, normalized by n/root log n, they converge in L-p to a constant for all p > 0. These results are compared with the extreme gaps betw...
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作者：Grimmett, Geoffrey R.; Manolescu, Ioan
作者单位：University of Cambridge
摘要：The star triangle transformation is used to obtain an equivalence extending over the set of all (in)homogeneous bond percolation models on the square, triangular and hexagonal lattices. Among the consequences are box-crossing (RSW) inequalities for such models with parameter-values at which the transformation is valid. This is a step toward proving the universality and conformality of these processes. It implies criticality of such values, thereby providing a new proof of the critical point of...
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作者：Hairer, Martin; Pillai, Natesh S.
作者单位：University of Warwick; Harvard University
摘要：We consider differential equations driven by rough paths and study the regularity of the laws and their long time behavior. In particular, we focus on the case when the driving noise is a rough path valued fractional Brownian motion with Hurst parameter H is an element of (1/3, 1/2]. Our contribution in this work is twofold. First, when the driving vector fields satisfy Hormander's celebrated Lie bracket condition, we derive explicit quantitative bounds on the inverse of the Malliavin matrix. ...
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作者：Nourdin, Ivan; Peccati, Giovanni
作者单位：Universite de Lorraine
摘要：We prove that an adequately resealed sequence {F-n} of self-adjoint operators, living inside a fixed free Wigner chaos of even order, converges in distribution to a centered free Poisson random variable with rate lambda > 0 if and only if phi(F-n(4)) - 2 phi(F-n(3)) -> 2 lambda(2) - lambda (where phi is the relevant tracial state). This extends to a free setting some recent limit theorems by Nourdin and Peccati [Ann. Probab. 37 (2009) 1412-1426] and provides a noncentral counterpart to a resul...
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作者：Bakhtin, Yuri
作者单位：University System of Georgia; Georgia Institute of Technology
摘要：We consider the Burgers equation on the real line with forcing given by Poissonian noise with no periodicity assumption. Under a weak concentration condition on the driving random force, we prove existence and uniqueness of a global solution in a certain class. We describe its basin of attraction that can also be viewed as the main ergodic component for the model. We establish existence and uniqueness of global minimizers associated to the variational principle underlying the dynamics. We also...
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作者：Le Gall, Jean-Francois
作者单位：Universite Paris Saclay
摘要：We consider a random planar map M-n which is uniformly distributed over the class of all rooted q-angulations with n faces. We let m(n) be the vertex set of M-n, which is equipped with the graph distance d(gr). Both when q >= 4 is an even integer and when q = 3, there exists a positive constant c(q) such that the rescaled metric spaces (m(n), c(q)n(-1/4)d(gr)) converge in distribution in the Gromov-Hausdorff sense, toward a universal limit called the Brownian map. The particular case of triang...
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作者：Cator, Eric; Pimentel, Leandro P. R.
作者单位：Delft University of Technology; Universidade Federal do Rio de Janeiro
摘要：In this paper we will show how the results found in [Probab. Theory Related Fields 154 (2012) 89-125], about the Busemann functions in last-passage percolation, can be used to calculate the asymptotic distribution of the speed of a single second class particle starting from an arbitrary deterministic configuration which has a rarefaction fan, in either the totally asymetric exclusion process or the Hammersley interacting particle process. The method will be to use the well-known last-passage p...
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作者：Jung, Paul; Markowsky, Greg
作者单位：University of Alabama System; University of Alabama Birmingham; Monash University
摘要：For a random walk defined for a doubly infinite sequence of times, we let the time parameter itself be an integer-valued process, and call the orginal process a random walk at random time. We find the scaling limit which generalizes the so-called iterated Brownian motion. Khoshnevisan and Lewis [Ann. Appl. Probab. 9 (1999) 629-667] suggested the existence of a form of measure-theoretic duality between iterated Brownian motion and a Brownian motion in random scenery. We show that a random walk ...
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作者：Burdzy, Krzysztof; Chen, Zhen-Qing
作者单位：University of Washington; University of Washington Seattle
摘要：In this paper, we show that reflecting Brownian motion in any bounded domain D can be approximated, as k -> infinity, by simple random walks on maximal connected subsets of (2(-k)Z(d)) boolean AND D whose filled-in interiors are inside of D.
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作者：Cass, Thomas; Litterer, Christian; Lyons, Terry
作者单位：Imperial College London; University of Oxford
摘要：We derive explicit tail-estimates for the Jacobian of the solution flow for stochastic differential equations driven by Gaussian rough paths. In particular, we deduce that the Jacobian has finite moments of all order for a wide class of Gaussian process including fractional Brownian motion with Hurst parameter H > 1/4. We remark on the relevance of such estimates to a number of significant open problems.
