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作者：Ding, Jian
作者单位：Stanford University
摘要：For a complete graph of size n, assign each edge an i.i.d. exponential variable with mean n. For lambda > 0, consider the length of the longest path whose average weight is at most lambda. It was shown by Aldous [Combin. Probab. Comput. 7 (1998) 1-10] that the length is of order log n for lambda < 1/e and of order n for lambda > 1/e. Aldous [Open problems (2003) Preprint] posed the question on detailed behavior at and near criticality 1/e. In particular, Aldous asked whether there exist scalin...
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作者：Aidekon, Elie; Hu, Yueyun; Zindy, Olivier
作者单位：Eindhoven University of Technology; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Sorbonne Universite; Universite Paris Cite; Universite Paris 13; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
摘要：Consider a branching random walk on the real line with a killing barrier at zero: starting from a nonnegative point, particles reproduce and move independently, but are killed when they touch the negative half-line. The population of the killed branching random walk dies out almost surely in both critical and subcritical cases, where by subcritical case we mean that the rightmost particle of the branching random walk without killing has a negative speed, and by critical case, when this speed i...
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作者：Baik, Jinho; Jenkins, Robert
作者单位：University of Michigan System; University of Michigan
摘要：The notion of r-crossing and r-nesting of a complete matching was introduced and a symmetry property was proved by Chen et al. [Trans. Amer. Math. Soc. 359 (2007) 1555-1575]. We consider random matchings of large size and study their maximal crossing and their maximal nesting. It is known that the marginal distribution of each of them converges to the GOE Tracy-Widom distribution. We show that the maximal crossing and the maximal nesting becomes independent asymptotically, and we evaluate the ...
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作者：Sen, Arnab; Virag, Balint
作者单位：University of Minnesota System; University of Minnesota Twin Cities; University of Toronto; University of Toronto
摘要：We show that the top eigenvalue of an n x n random symmetric Toeplitz matrix, scaled by v root 2n log n, converges to the square of the 2 -> 4 operator norm of the sine kernel.
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作者：Basdevant, Anne-Laure; Enriquez, Nathanael; Gerin, Lucas
作者单位：Universite Paris Saclay
摘要：On the supercritical percolation cluster with parameter p, the distances between two distant points of the axis are asymptotically increased by a factor 1 + 1-p/2 + o(1 - p) with respect to the usual distance. The proof is based on an apparently new connection with the TASEP (totally asymmetric simple exclusion process).
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作者：Cerf, Raphael; Manzo, Francesco
作者单位：Universite Paris Saclay; Roma Tre University
摘要：This work extends to dimension d >= 3 the main result of Dehghanpour and Schonmann. We consider the stochastic Ising model on Z(d) evolving with the Metropolis dynamics under a fixed small positive magnetic field h starting from the minus phase. When the inverse temperature beta goes to infinity, the relaxation time of the system, defined as the time when the plus phase has invaded the origin, behaves like exp(beta k(d)). The value k(d) is equal to k(d) =1/d+1(Gamma(1) +...+ Gamma(d)), where G...
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作者：Bayati, Mohsen; Gamarnik, David; Tetali, Prasad
作者单位：Stanford University; Massachusetts Institute of Technology (MIT); University System of Georgia; Georgia Institute of Technology
摘要：We establish the existence of free energy limits for several combinatorial models on Erdos-Renyi graph G(N, [cN]) and random r-regular graph G(N, r). For a variety of models, including independent sets, MAX-CUT, coloring and K-SAT, we prove that the free energy both at a positive and zero temperature, appropriately rescaled, converges to a limit as the size of the underlying graph diverges to infinity. In the zero temperature case, this is interpreted as the existence of the scaling limit for ...
