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作者：Bordenave, Charles; Lelarge, Marc; Massoulie, Laurent
作者单位：Centre National de la Recherche Scientifique (CNRS); Universite de Toulouse; Universite Toulouse III - Paul Sabatier; Inria; Universite PSL; Ecole Normale Superieure (ENS); Institut Polytechnique de Paris; Ecole Polytechnique; Universite de Toulouse; Universite Toulouse III - Paul Sabatier; Centre National de la Recherche Scientifique (CNRS); Universite de Toulouse; Universite Toulouse III - Paul Sabatier; Inria
摘要：A nonbacktracking walk on a graph is a directed path such that no edge is the inverse of its preceding edge. The nonbacktracking matrix of a graph is indexed by its directed edges and can be used to count nonbacktracking walks of a given length. It has been used recently in the context of community detection and has appeared previously in connection with the Ihara zeta function and in some generalizations of Ramanujan graphs. In this work, we study the largest eigenvalues of the nonbacktrackin...
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作者：Cook, Nicholas; Goldstein, Larry; Johnson, Tobias
作者单位：Stanford University; University of Southern California
摘要：Let lambda be the second largest eigenvalue in absolute value of a uniform random d-regular graph on n vertices. It was famously conjectured by Alon and proved by Friedman that if d is fixed independent of n, then lambda = 2 root d - 1+ o(1) with high probability. In the present work, we show that lambda = O(root d) continues to hold with high probability as long as d = O(n(2/3)), making progress toward a conjecture of Vu that the bound holds for all 1 <= d <= n/2. Prior to this work the best ...
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作者：Berestycki, Nathanael; Lubetzky, Eyal; Peres, Yuval; Sly, Allan
作者单位：University of Cambridge; New York University; Microsoft; University of California System; University of California Berkeley
摘要：We study random walks on the giant component of the Erdos-Renyi random graph G(n, p) where p = lambda/n for lambda > 1 fixed. The mixing time from a worst starting point was shown by Fountoulakis and Reed, and independently by Benjamini, Kozma and Wormald, to have order log(2) n. We prove that starting from a uniform vertex (equivalently, from a fixed vertex conditioned to belong to the giant) both accelerates mixing to O(log n) and concentrates it (the cutoff phenomenon occurs): the typical m...
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作者：Andres, Sebastian; Chiarini, Alberto; Deuschel, Jean-Dominique; Slowik, Martin
作者单位：University of Cambridge; Aix-Marseille Universite; Technical University of Berlin
摘要：We study a continuous-time random walk, X, on Z(d) in an environment of dynamic random conductances taking values in (0,infinity). We assume that the law of the conductances is ergodic with respect to space-time shifts. We prove a quenched invariance principle for the Markov process X under some moment conditions on the environment. The key result on the sublinearity of the corrector is obtained by Moser's iteration scheme.
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作者：Cosso, Andrea; Federico, Salvatore; Gozzi, Fausto; Rosestolato, Mauro; Touzi, Nizar
作者单位：Polytechnic University of Milan; University of Siena; Luiss Guido Carli University; Luiss Guido Carli University; Institut Polytechnique de Paris; Ecole Polytechnique
摘要：Path-dependent partial differential equations (PPDEs) are natural objects to study when one deals with non-Markovian models. Recently, after the introduction of the so-called pathwise (or functional or Dupire) calculus [see Dupire (2009)], in the case of finite-dimensional underlying space various papers have been devoted to studying the well-posedness of such kind of equations, both from the point of view of regular solutions [see, e.g., Dupire (2009) and Cont (2016) Stochastic Integration by...
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作者：Bolley, Francois; Gentil, Ivan; Guillin, Arnaud
作者单位：Universite Paris Cite; Sorbonne Universite; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Ecole Centrale de Lyon; Institut National des Sciences Appliquees de Lyon - INSA Lyon; Universite Claude Bernard Lyon 1; Universite Jean Monnet; Universite Clermont Auvergne (UCA); Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
摘要：In this work, we consider dimensional improvements of the logarithmic Sobolev, Talagrand and Brascamp-Lieb inequalities. For this, we use optimal transport methods and the Borell-Brascamp-Lieb inequality. These refinements can be written as a deficit in the classical inequalities. They have the right scale with respect to the dimension. They lead to sharpened concentration properties as well as refined contraction bounds, convergence to equilibrium and short time behavior for the laws of solut...
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作者：Zhu, Rongchan; Zhu, Xiangchan
作者单位：Beijing Institute of Technology; Beijing Jiaotong University; University of Bielefeld
摘要：We study the lattice approximations to the dynamical Phi(4)(3) model by paracontrolled distributions proposed in [Forum Math. Pi 3 (2015) e6]. We prove that the solutions to the lattice systems converge to the solution to the dynamical Phi(4)(3) model in probability, locally uniformly in time. Since the dynamical Phi(4)(3) model is not well defined in the classical sense and renormalisation has to be performed in order to define the nonlinear term, a corresponding suitable drift term is added ...
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作者：Borgs, Christian; Chayes, Jennifer T.; Cohn, Henry; Zhao, Yufei
作者单位：Microsoft; Massachusetts Institute of Technology (MIT)
摘要：We extend the L-p theory of sparse graph limits, which was introduced in a companion paper, by analyzing different notions of convergence. Under suitable restrictions on node weights, we prove the equivalence of metric convergence, quotient convergence, microcanonical ground state energy convergence, microcanonical free energy convergence and large deviation convergence. Our theorems extend the broad applicability of dense graph convergence to all sparse graphs with unbounded average degree, w...
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作者：Hammond, Alan
作者单位：University of California System; University of California Berkeley; University of California System; University of California Berkeley
摘要：For d >= 2 and n is an element of N even, let p(n) = p(n)(d) denote the number of length n self-avoiding polygons in Z(d) up to translation. The polygon cardinality grows exponentially, and the growth rate lim(n is an element of 2N) (1/n)(pn) is an element of (0,infinity) is called the connective constant and denoted by mu. Madras [J. Stat. Phys. 78 (1995) 681-699] has shown that p(n)mu(-n) <= Cn(-1/2) in dimension d = 2. Here, we establish that p(n)mu(-n) <= n(-3/2+o(1)) for a set of even n o...
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作者：Bertoin, Jean; Curien, Nicolas; Kortchemski, Igor
作者单位：University of Zurich; Universite Paris Saclay; Centre National de la Recherche Scientifique (CNRS); Institut Polytechnique de Paris; Ecole Polytechnique
摘要：We are interested in the cycles obtained by slicing at all heights random Boltzmann triangulations with a simple boundary. We establish a functional invariance principle for the lengths of these cycles, appropriately rescaled, as the size of the boundary grows. The limiting process is described using a self-similar growth-fragmentation process with explicit parameters. To this end, we introduce a branching peeling exploration of Boltzmann triangulations, which allows us to identify a crucial m...
