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作者：Damron, Michael; Hanson, Jack; Lam, Wai-Kit
作者单位：University System of Georgia; Georgia Institute of Technology; City University of New York (CUNY) System; City College of New York (CUNY); Indiana University System; Indiana University Bloomington
摘要：First-passage percolation is a random growth model defined using i.i.d. edge-weights (t(e)) on the nearest-neighbor edges of Z(d). An initial infection occupies the origin and spreads along the edges, taking time t(e) to cross the edge e. In this paper, we study the size of the boundary of the infected (wet) region at time t, B(t). It is known that B(t) grows linearly, so its boundary partial derivative B(t) has size between ct(d-1) and Ct(d). Under a weak moment condition on the weights, we s...
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作者：Kiwi, Marcos; Mitsche, Dieter
作者单位：Universidad de Chile; Universidad de Chile; Universite Cote d'Azur
摘要：Random hyperbolic graphs have been suggested as a promising model of social networks. A few of their fundamental parameters have been studied. However, none of them concerns their spectra. We consider the random hyperbolic graph model, as formalized by [Automata, Languages, and Programming-39th International Colloquium-ICALP Part II. (2012) 573-585 Springer], and essentially determine the spectral gap of their normalized Laplacian. Specifically, we establish that with high probability the seco...
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作者：Louart, Cosme; Liao, Zhenyu; Couillet, Romain
作者单位：Universite Paris Saclay
摘要：This article studies the Gram random matrix model G = 1/T Sigma E-T E = sigma (WX), classically found in the analysis of random feature maps and random neural networks, where X = [x(1),, x(T)] epsilon R-PXT is a (data) matrix of bounded norm, W epsilon R-nxp) is a matrix of independent zero-mean unit variance entries and o : R -> R is a Lipschitz continuous (activation) function-sigma (WX) being understood entry-wise. By means of a key concentration of measure lemma arising from nonasymptotic ...
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作者：Giacomin, Giambattista; Lacoin, Hubert
作者单位：Sorbonne Universite; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite Paris Cite; Instituto Nacional de Matematica Pura e Aplicada (IMPA)
摘要：We consider the lattice Gaussian free field in d + 1 dimensions, d = 3 or larger, on a large box (linear size N) with boundary conditions zero. On this field, two potentials are acting: one, that models the presence of a wall, penalizes the field when it enters the lower half space and one, the pinning potential, rewards visits to the proximity of the wall. The wall can be soft, that is, the field has a finite penalty to enter the lower half-plane, or hard, when the penalty is infinite. In gen...
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作者：Hening, Alexandru; Nguyen, Dang H.
作者单位：Tufts University; Wayne State University
摘要：In recent years there has been a growing interest in the study of the dynamics of stochastic populations. A key question in population biology is to understand the conditions under which populations coexist or go extinct. Theoretical and empirical studies have shown that coexistence can be facilitated or negated by both biotic interactions and environmental fluctuations. We study the dynamics of n populations that live in a stochastic environment and which can interact nonlinearly (through com...
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作者：Fountoulakis, Nikolaos; Kang, Mihyun; Koch, Christoph; Makai, Tamas
作者单位：University of Birmingham; Graz University of Technology; University of Warwick
摘要：A bootstrap percolation process on a graph with infection threshold r >= 1 is a dissemination process that evolves in time steps. The process begins with a subset of infected vertices and in each subsequent step every uninfected vertex that has at least r infected neighbours becomes infected and remains so forever. Critical phenomena in bootstrap percolation processes were originally observed by Aizenman and Lebowitz in the late 1980s as finite-volume phase transitions in Z(d) that are caused ...
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作者：Hiraoka, Yasuaki; Shirai, Tomoyuki; Khanh Duy Trinh
作者单位：Tohoku University; Kyushu University
摘要：The persistent homology of a stationary point process on R-N is studied in this paper. As a generalization of continuum percolation theory, we study higher dimensional topological features of the point process such as loops, cavities, etc. in a multiscale way. The key ingredient is the persistence diagram, which is an expression of the persistent homology. We prove the strong law of large numbers for persistence diagrams as the window size tends to infinity and give a sufficient condition for ...
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作者：Forman, Noah; Pal, Soumik; Rizzolo, Douglas; Winkel, Matthias
作者单位：University of Washington; University of Washington Seattle; University of Delaware; University of Oxford
摘要：We establish two results about local times of spectrally positive stable processes. The first is a general approximation result, uniform in space and on compact time intervals, in a model where each jump of the stable process may be marked by a random path. The second gives moment control on the Holder constant of the local times, uniformly across a compact spatial interval and in certain random time intervals. For the latter, we introduce the notion of a Levy process restricted to a compact i...
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作者：Roberts, Matthew, I; Sengul, Bati
作者单位：University of Bath; Bank of America Corporation
摘要：In this paper, we introduce a network model which evolves in time, and study its largest connected component. We consider a process of graphs (G(t) : t is an element of [0, 1]), where initially we start with a critical Erdos-Renyi graph ER(n, 1/n), and then evolve forward in time by resampling each edge independently at rate 1. We show that the size of the largest connected component that appears during the time interval [0, 1] is of order n(2/3) log(1/3)n with high probability. This is in con...
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作者：Mendelson, Shahar; Rauhut, Holger; Ward, Rachel
作者单位：Technion Israel Institute of Technology; Australian National University; RWTH Aachen University; University of Texas System; University of Texas Austin
摘要：We study the recovery of sparse vectors from subsampled random convolutions via l(1)-minimization. We consider the setup in which both the subsampling locations as well as the generating vector are chosen at random. For a sub-Gaussian generator with independent entries, we improve previously known estimates: if the sparsity s is small enough, that is, s less than or similar to root n/log(n), we show that m greater than or similar to s log(en/s) measurements are sufficient to recover s-sparse v...
