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作者：Butkovsky, Oleg; Mytnik, Leonid
作者单位：Technion Israel Institute of Technology
摘要：Motivated by the regularization by noise phenomenon for SDEs, we prove existence and uniqueness of the flow of solutions for the non-Lipschitz stochastic heat equation partial derivative u/partial derivative t = 1/2 partial derivative(2)u/partial derivative z(2) + b(u(t,z)) + <(W)single over dot>(t,z), where <(W)single over dot> is a space-time white noise on R+ x R and b is a bounded measurable function on R. As a byproduct of our proof, we also establish the so-called path-by-path uniqueness...
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作者：Dalang, Robert C.; Khoshnevisan, Davar; Zhang, Tusheng
作者单位：Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne; Utah System of Higher Education; University of Utah; University of Manchester
摘要：Let xi(t, x) denote space-time white noise and consider a reaction-diffusion equation of the form <(u) over dot>(t, x) = 1/2u ''(t, x) + b(u(t, x)) + sigma(u(t,x))xi(t,x) on R+ x [0, 1], with homogeneous Dirichlet boundary conditions and suitable initial data, in the case that there exists epsilon > 0 such that vertical bar b(z)vertical bar >= vertical bar z vertical bar (log vertical bar z vertical bar)(1+epsilon) for all sufficiently-large values of vertical bar z vertical bar. When sigma eq...
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作者：Bauerschmidt, Roland; Helmuth, Tyler; Swan, Andrew
作者单位：University of Cambridge; University of Bristol
摘要：We prove the vertex-reinforced jump process (VRJP) is recurrent in two dimensions for any translation invariant finite-range initial rates. Our proof has two main ingredients. The first is a direct connection between the VRJP and sigma models whose target space is a hyperbolic space H-n or its super-symmetric counterpart H-2 broken vertical bar 2. These results are analogues of well-known relations between the Gaussian free field and the local times of simple random walk. The second ingredient...
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作者：Fan, Jianqing; Jiang, Tiefeng
作者单位：Princeton University; University of Minnesota System; University of Minnesota Twin Cities
摘要：The paper studies the limiting distribution of the largest off-diagonal entry of the sample correlation matrices of high-dimensional Gaussian populations with equi-correlation structure. Assume the entries of the population distribution have a common correlation coefficient rho > 0 and both the population dimension p and the sample size n tend to infinity with log p = o(n(1/3)). As 0 < rho < 1, we prove that the largest off-diagonal entry of the sample correlation matrix converges to a Gaussia...
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作者：Balazs, Marton; Rassoul-Agha, Firas; Seppalainen, Timo
作者单位：University of Bristol; Utah System of Higher Education; University of Utah; University of Wisconsin System; University of Wisconsin Madison
摘要：Random walk in a dynamic i.i.d. beta random environment, conditioned to escape at an atypical velocity, converges to a Doob transform of the original walk. The Doob-transformed environment is correlated in time, i.i.d. in space and its marginal density function is a product of a beta density and a hypergeometric function. Under its averaged distribution, the transformed walk obeys the wandering exponent 2/3 that agrees with Kardar-Parisi-Zhang universality. The harmonic function in the Doob tr...
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作者：Kenyon, Richard; Miller, Jason; Sheffield, Scott; Wilson, David B.
作者单位：Brown University; University of Cambridge; Massachusetts Institute of Technology (MIT); University of Washington; University of Washington Seattle
摘要：We give bijections between bipolar-oriented (acyclic with unique source and sink) planar maps and certain random walks, which show that the uniformly random bipolar-oriented planar map, decorated by the peano curve surrounding the tree of left-most paths to the sink, converges in law with respect to the peanosphere topology to a root 4/3-Liouville quantum gravity surface decorated by an independent Schramm-Loewner evolution with parameter kappa = 12 (i.e., SLE12). This result is universal in t...
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作者：Sznitman, Alain-Sol
作者单位：Swiss Federal Institutes of Technology Domain; ETH Zurich
摘要：We consider Z(d), d >= 3. We investigate the vacant set V-u of random interlacements in the strongly percolative regime, the vacant set V of the simple random walk and the excursion set E->=alpha of the Gaussian free field in the strongly percolative regime. We consider the large deviation probability that the adequately thickened component of the boundary of a large box centered at the origin in the respective vacant sets or excursion set leaves in the box a macroscopic volume in its compleme...
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作者：Gerencser, Mate
作者单位：Institute of Science & Technology - Austria
摘要：The boundary behaviour of solutions of stochastic PDEs with Dirichlet boundary conditions can be surprisingly-and in a sense, arbitrarily-bad: as shown by Krylov [SIAM J. Math. Anal. 34 (2003) 1167-1182], for any alpha > 0 one can find a simple 1-dimensional constant coefficient linear equation whose solution at the boundary is not alpha-Holder continuous. We obtain a positive counterpart of this: under some mild regularity assumptions on the coefficients, solutions of semilinear SPDEs on C-1 ...
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作者：Berger, Quentin; Torri, Niccolo
作者单位：Sorbonne Universite
摘要：We study the directed polymer model in dimension 1 + 1 when the environment is heavy-tailed, with a decay exponent alpha is an element of (0, 2). We give all possible scaling limits of the model in the weak-coupling regime, that is, when the inverse temperature temperature beta = beta(n) vanishes as the size of the system n goes to infinity. When alpha is an element of (1/2, 2), we show that all possible transversal fluctuations root n <= h(n) <= n can be achieved by tuning properly beta(n), a...
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作者：He, Yukun; Knowles, Antti; Marcozzi, Matteo
作者单位：University of Zurich; University of Geneva
摘要：We prove a local law for the adjacency matrix of the Erdos-Renyi graph G(N, p) in the supercritical regime pN >= C logN where G(N, p) has with high probability no isolated vertices. In the same regime, we also prove the complete delocalization of the eigenvectors. Both results are false in the complementary subcritical regime. Our result improves the corresponding results from (Ann. Probab. 41 (2013) 2279-2375) by extending them all the way down to the critical scale pN = O(logN). A key ingred...
