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作者：Erde, J. O. S. H. U. A.; Kang, M. I. H. Y. U. N.; Krivelevich, M. I. C. H. A. E. L.
作者单位：Graz University of Technology; Tel Aviv University
摘要：It is well known that the behaviour of a random subgraph of a d-dimensional hypercube, where we include each edge independently with probability p, undergoes a phase transition when p is around 1/d More precisely, standard arguments show that just below this value of p all components of this graph have order O(d) with probability tending to one as d -> infinity (whp for short), whereas Ajtai, Komlos and Szemeredi (Combinatorica 2 (1982) 1-7) showed that just above this value, in the supercriti...
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作者：Alimohammadi, Yeganeh; Borgs, Christian; Saberi, Amin
作者单位：Stanford University; University of California System; University of California Berkeley
摘要：We study random digraphs on sequences of expanders with a bounded average degree which converge locally in probability. We prove that the rel-ative size and the threshold for the existence of a giant strongly connected component as well as the asymptotic fraction of nodes with giant fan-in or nodes with giant fan-out are local, in the sense that they are the same for two sequences with the same local limit. The digraph has a bow-tie structure, with all but a vanishing fraction of nodes lying e...
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作者：Alishahi, Kasra; Barzegar, Milad; Zamani, Mohammadsadegh
作者单位：Sharif University of Technology; University of Yazd
摘要：We prove that every negatively associated sequence of Bernoulli random variables with summable covariances has a trivial tail a-field. A corollary of this result is the tail triviality of strongly Rayleigh processes. This is a gen-eralization of a result due to Lyons, which establishes tail triviality for dis-crete determinantal processes. We also study the tail behavior of negatively associated Gaussian and Gaussian threshold processes. We show that these processes are tail trivial though, in...
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作者：Dumaz, Laure; Labbe, Cyril
作者单位：Universite PSL; Ecole Normale Superieure (ENS); Centre National de la Recherche Scientifique (CNRS); Universite Paris Cite; Universite Paris Cite
摘要：We introduce a random differential operator that we call CS tau operator, whose spectrum is given by the Sch(tau) point process introduced by Kritchevski, Valko and Virag (Comm. Math Phys. (2012) 314 775-806) and whose eigen-vectors match with the description provided by Rifkind and Virag (Geom. Funct. Anal. (2018) 28 1394-1419). This operator acts on R-2-valued func-tions from the interval [0, 1] and takes the form [GRAPHICS] . where dB, dW(1) and dW(2) are independent white noises. Then we i...
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作者：Chen, Xin; LI, Xue-mei; Wu, Bo
作者单位：Shanghai Jiao Tong University; Imperial College London; Fudan University
摘要：The main results of the article are short time estimates and asymptotic estimates for the first two order derivatives of the logarithmic heat kernel of a complete Riemannian manifold. We remove all curvature restrictions and also develop several techniques.A basic tool developed here is intrinsic stochastic variations with pre-scribed second order covariant differentials, allowing to obtain a path inte-gration representation for the second order derivatives of the heat semigroup Pt on a comple...
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作者：Schmid, Dominik
作者单位：University of Bonn
摘要：We study mixing times for the totally asymmetric simple exclusion process (TASEP) on a segment of size N with open boundaries. We focus on the maximal current phase and prove that the mixing time is of order N3/2, up to logarithmic corrections. In the triple point, where the TASEP with open boundaries approaches the Uniform distribution on the state space, we show that the mixing time is precisely of order N3/2. This is conjectured to be the correct order of the mixing time for a wide range of...
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作者：Hutchroft, Tom; Michta, Emmanue; Slade, Gordon
作者单位：California Institute of Technology; University of British Columbia
摘要：We consider percolation on Z(d) and on the d-dimensional discrete torus, in dimensions d >= 11 for the nearest-neighbour model and in dimensions d > 6 for spread-out models. For Z(d) we employ a wide range of techniques and previous results to prove that there exist positive constants c and C such that the slightly subcritical two-point function and one-arm probabilities satisfy Ppc-epsilon(0 <-> x) <= C/||x||(d-2) e(-c epsilon 1/2) ||x|| c/r(2)e -C epsilon(1/2r) <= Ppc-epsilon (0 <-> partial ...
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作者：Duminil-Copin, Hugo; Rivera, Alejandro; Rodriguez, Pierre-Francois; Vanneuville, Hugo
作者单位：University of Geneva; Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne; Imperial College London; Communaute Universite Grenoble Alpes; Universite Grenoble Alpes (UGA)
摘要：For the Bargmann-Fock field on R-d with d >= 3, we prove that the critical level l(c) (d) of the percolation model formed by the excursion sets {f >= l} is strictly positive. This implies that for every l sufficiently close to 0 (in particular for the nodal hypersurfaces corresponding to the case l = 0), {f = l} contains an unbounded connected component that visits most of the ambient space. Our findings actually hold for a more general class of positively correlated smooth Gaussian fields wit...
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作者：Addario-Berry, Louigi; Barrett, Jordan
作者单位：McGill University
摘要：We consider invasion percolation on the complete graph K-n, started from some number k(n) of distinct source vertices. The outcome of the process is a forest consisting of k(n) trees, each containing exactly one source. Let M-n be the size of the largest tree in this forest. Logan, Molloy and Pralat (2018) proved that if k(n)/n(1/3)-> 0 then M-n/n -> 1 in probability. In this paper, we prove a complementary result: if k(n)/n(1/3)-> infinity, then M-n/n -> 0 in probability. This establishes the...
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作者：Fox, Jacob; Kwan, Matthew; Spink, Hunter
作者单位：Stanford University
摘要：The classical Erdos-Littlewood-Offord theorem says that for nonzero vectors a1, ... , an E Rd, any x E Rd, and uniformly random (xi 1,. . . ,xi n) E {-1, 1}n, we have Pr(a1 xi 1 + middot middot middot + an xi n = x) = O(n-1/2). In this paper, we show that Pr(a1 xi 1 + middot middot middot + an xi n E S) < n-1/2+o(1) whenever S is definable with respect to an o-minimal structure (e.g., this holds when S is any algebraic hypersurface), under the necessary condition that it does not contain a lin...
