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作者：Biswas, Ratul; Chen, Wei-kuo; Sen, Arnab
作者单位：University of Minnesota System; University of Minnesota Twin Cities
摘要：We study a diluted mean-field spin glass model with a quadratic Hamil-tonian. Our main result establishes the limiting free energy in terms of an in-tegral of a family of random variables that are the weak limits of the quenched variances of the spins in the system with varying edge connectivity. The key ingredient in our argument is played by the identification of these random variables as the unique solution to a recursive distributional equation. Our re-sults in particular provide the first...
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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：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：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...
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作者：Conchon-Kerjan, Guillaume; Goldschmidt, Christina
作者单位：University of Bath; University of Oxford
摘要：We prove a metric space scaling limit for a critical random graph with independent and identically distributed degrees having power-law tail be-haviour with exponent alpha + 1, where alpha & ISIN; (1, 2). The limiting components are constructed from random R-trees encoded by the excursions above its run-ning infimum of a process whose law is locally absolutely continuous with respect to that of a spectrally positive alpha-stable Levy process. These spanning R-trees are measure-changed alpha-st...
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作者：Landim, C.; Marcondes, D.; Seo, I
作者单位：Instituto Nacional de Matematica Pura e Aplicada (IMPA); Universite de Rouen Normandie; Universidade de Sao Paulo; Seoul National University (SNU); Seoul National University (SNU)
摘要：We present a general method to derive the metastable behavior of weakly mixing Markov chains. This approach is based on properties of the resolvent equations and can be applied to metastable dynamics, which do not satisfy the As an application, we study the metastable behavior of critical zero-range processes. Let r : S x S -> R+ be the jump rates of an irreducible random walk on a finite set S, reversible with respect to the uniform measure. For alpha > 0, let g : N -> R+ be given by g(0) = 0...
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作者：Alt, Johannes; Ducatez, Raphael; Knowles, Antti
作者单位：University of Geneva; New York University; Ecole Normale Superieure de Lyon (ENS de LYON)
摘要：We consider the adjacency matrix A of the Erdos-Renyi graph on N ver-tices with edge probability d/N. For (log log N)4 << d < log N, we prove that the eigenvalues near the spectral edge form asymptotically a Poisson point process and the associated eigenvectors are exponentially localized. As a corollary, at the critical scale d kappa log N, the limiting distribution of the largest nontrivial eigenvalue does not match with any previously known dis-tribution. Together with (Comm. Math. Phys. 38...
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作者：Brito, Gerandy; Damron, Michael; Hanson, Jack
作者单位：University System of Georgia; Georgia Institute of Technology; University System of Georgia; Georgia Institute of Technology; City University of New York (CUNY) System; City College of New York (CUNY)
摘要：In first-passage percolation (FPP), one places nonnegative random vari-ables (weights) (te) on the edges of a graph and studies the induced weighted graph metric. We consider FPP on Zd for d > 2 and analyze the geometric properties of geodesics, which are optimizing paths for the metric. Specifi-cally, we address the question of existence of bigeodesics, which are doubly-infinite paths whose subpaths are geodesics. It is a famous conjecture origi-nating from a question of Furstenberg and most ...
