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作者：Panchenko, Dmitry
作者单位：Texas A&M University System; Texas A&M University College Station
摘要：The Parisi formula for the free energy in the Sherrington-Kirkpatrick and mixed p-spin models for even p >= 2 was proved in the seminal work of Michel Talagrand [Ann. of Math. (2) 163 (2006) 221-263]. In this paper we prove the Parisi formula for general mixed p-spin models which also include p-spin interactions for odd p. Most of the ideas used in the paper are well known and can now be combined following a recent proof of the Parisi ultrametricity conjecture in.
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作者：Basdevant, Anne-Laure; Schapira, Bruno; Singh, Arvind
作者单位：Universite Paris Nanterre; Universite Paris Saclay; Universite Paris Saclay
摘要：We characterize nondecreasing weight functions for which the associated one-dimensional vertex reinforced random walk (VRRW) localizes on 4 sites. A phase transition appears for weights of order n log log n: for weights growing faster than this rate, the VRRW localizes almost surely on, at most, 4 sites, whereas for weights growing slower, the VRRW cannot localize on less than 5 sites. When w is of order n log log n, the VRRW localizes almost surely on either 4 or 5 sites, both events happenin...
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作者：Cuny, Christophe; Merlevede, Florence
作者单位：Universite Paris Saclay; Universite Paris-Est-Creteil-Val-de-Marne (UPEC); Universite Gustave-Eiffel; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
摘要：In this paper, we obtain sufficient conditions in terms of projective criteria under which the partial sums of a stationary process with values in H (a real and separable Hilbert space) admits an approximation, in L-p (H), p > 1, by a martingale with stationary differences, and we then estimate the error of approximation in L-p (H). The results are exploited to further investigate the behavior of the partial sums. In particular we obtain new projective conditions concerning the Marcinkiewicz-Z...
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作者：Einmahl, Uwe; Kuelbs, Jim
作者单位：Vrije Universiteit Brussel; University of Wisconsin System; University of Wisconsin Madison
摘要：Let X, X-1, X-2,... be i.i.d, mean zero random vectors with values in a separable Banach space B, S-n = X-1 +...+ X-n for n >= 1, and assume {c(n) : n >= 1} is a suitably regular sequence of constants. Furthermore, let S((n)) (t), 0 <= t <= 1 be the corresponding linearly interpolated partial sum processes. We study the cluster sets A = C({S-n/c(n)}) and A = C({S((n))(.)/c(n)}). In particular, A and A are shown to be nonrandom, and we derive criteria when elements in B and continuous functions...
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作者：Castell, Fabienne; Guillotin-Plantard, Nadine; Pene, Francoise; Schapira, Bruno
作者单位：Aix-Marseille Universite; Centre National de la Recherche Scientifique (CNRS); Ecole Centrale de Lyon; Institut National des Sciences Appliquees de Lyon - INSA Lyon; Universite Claude Bernard Lyon 1; Universite Jean Monnet; CNRS - National Institute for Mathematical Sciences (INSMI); Universite de Bretagne Occidentale; Centre National de la Recherche Scientifique (CNRS); Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite Paris Saclay
摘要：Random walks in random scenery are processes defined by Z(n) := Sigma(n)(k=1) xi(X1+ ...+Xk), where basically (X-k, k >= 1) and (xi(y), y is an element of Z) are two independent sequences of i.i.d. random variables. We assume here that X-1 is Z-valued, centered and with finite moments of all orders. We also assume that xi(0) is Z-valued, centered and square integrable. In this case H. Kesten and F. Spitzer proved that (n(-3/4)Z([nt]), t >= 0) converges in distribution as n --> infinity toward ...
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作者：Wang, Feng-Yu
作者单位：Beijing Normal University; Swansea University
摘要：A new coupling argument is introduced to establish Driver's integration by parts formula and shift Harnack inequality. Unlike known coupling methods where two marginal processes with different starting points are constructed to move together as soon as possible, for the new-type coupling the two marginal processes start from the same point but their difference is aimed to reach a fixed quantity at a given time. Besides the integration by parts formula, the new coupling method is also efficient...
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作者：Goetze, Friedrich; Venker, Martin
作者单位：University of Bielefeld
摘要：We study local correlations of certain interacting particle systems on the real line which show repulsion similar to eigenvalues of random Hermitian matrices. Although the new particle system does not seem to have a natural spectral or determinantal representation, the local correlations in the bulk coincide in the limit of infinitely many particles with those known from random Hermitian matrices; in particular they can be expressed as determinants of the so-called sine kernel. These results m...
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作者：Sly, Allan; Sun, Nike
作者单位：University of California System; University of California Berkeley; Australian National University; Stanford University
摘要：We establish that the normalized log-partition function of any two-spin system on bipartite locally tree-like graphs converges to a limiting free energy density which coincides with the (nonrigorous) Bethe prediction of statistical physics. Using this result, we characterize the local structure of two-spin systems on locally tree-like bipartite expander graphs without the use of the second moment method employed in previous works on these questions. As a consequence, we show that for both the ...
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作者：Bollobas, Bela; Holmgren, Cecilia; Smith, Paul; Uzzell, Andrew J.
作者单位：University of Cambridge; University of Memphis
摘要：Let r is an element of N. In r-neighbour bootstrap percolation on the vertex set of a graph G, vertices are initially infected independently with some probability p. At each time step, the infected set expands by infecting all uninfected vertices that have at least r infected neighbours. When p is close to 1, we study the distribution of the time at which all vertices become infected. Given t = t (n) = o (log n/ log log n), we prove a sharp threshold result for the probability that percolation...
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作者：Cerf, Raphael; Theret, Marie
作者单位：Universite Paris Saclay; Universite Paris Saclay; Universite Paris Cite
摘要：We consider the standard first passage percolation model in the resealed graph Z(d)/n for d >= 2 and a domain Omega of boundary Gamma in R-d Let Gamma(1) and Gamma(2) be two disjoint open subsets of Gamma, representing the parts of Gamma through which some water can enter and escape from Omega. A law of large numbers for the maximal flow from Gamma(1), to Gamma(2) in Omega is already known. In this paper we investigate the asymptotic behavior of a maximal stream and a minimal cutset. A maximal...
