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作者：Yang, Jun; Rosenthal, Jeffrey S.
作者单位：University of Oxford; University of Toronto
摘要：This paper considers how to obtain MCMC quantitative convergence bounds which can be translated into tight complexity bounds in high-dimensional settings. We propose a modified drift-and-minorization ap-proach, which establishes generalized drift conditions defined in subsets of the state space. The subsets are called the large sets, and are chosen to rule out some bad states which have poor drift property when the dimension of the state space gets large. Using the large sets together with a f...
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作者：Alsmeyer, Gerold; Brofferio, Sara; Buraczewski, Dariusz
作者单位：University of Munster; Centre National de la Recherche Scientifique (CNRS); Universite Paris Saclay; University of Wroclaw
摘要：Given a sequence of i.i.d. random functions Psi(n) : R -> R, n is an element of N, we consider the iterated function system and Markov chain, which is recursively defined by X-0(x) := x and X-n(x) := Psi(n-1)(X-n-1(x)) for x is an element of R and n is an element of N. Under the two basic assumptions that the Psi(n) are a.s. continuous at any point in R and asymptotically linear at the endpoints +/-infinity, we study the tail behavior of the stationary laws of such Markov chains by means of Ma...
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作者：Schulte, Matthias; Yukich, J. E.
作者单位：Hamburg University of Technology; Lehigh University
摘要：We employ stabilization methods and second order Poincare inequalities to establish rates of multivariate normal convergence for a large class of vec-tors (H(1) s ,..., H(m) s ), s >= 1, of statistics of marked Poisson processes on Rd, d >= 2, as the intensity parameters tends to infinity. Our results are appli-cable whenever the functionals H(i) s , i is an element of {1, ... , m}, are expressible as sums of exponentially stabilizing score functions satisfying a moment condition. The rates ar...
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作者：Hanin, Boris
作者单位：Princeton University
摘要：This article gives a new proof that fully connected neural networks with random weights and biases converge to Gaussian processes in the regime where the input dimension, output dimension, and depth are kept fixed, while the hidden layer widths tend to infinity. Unlike prior work, convergence is shown assuming only moment conditions for the distribution of weights and for quite general nonlinearities.
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作者：Johnston, Samuel G. G.; Kyprianou, Andreas; Rogers, Tim
作者单位：University of London; King's College London; University of Warwick; University of Bath
摘要：Consider a multitype coalescent process in which each block has a colour in {1, ... , d}. Individual blocks may change colour, and some number of blocks of various colours may merge to form a new block of some colour. We show that if the law of a multitype coalescent process is invariant under permutations of blocks of the same colour, has consistent Markovian projec-tions, and has asynchronous mergers, then it is a multitype A-coalescent: a process in which single blocks may change colour, tw...
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作者：Peltola, Eveliina; Wu, Hao
作者单位：Aalto University; Tsinghua University
摘要：We prove that in the scaling limit, the crossing probabilities of multiple interfaces in the critical planar Ising model with alternating boundary condi-tions are conformally invariant expressions given by the pure partition func-tions of multiple SLE & kappa; with & kappa; = 3. In particular, this identifies the scaling limits with ratios of specific correlation functions of conformal field theory.
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作者：Alimohammadi, Yeganeh; Diaconis, Persi; Roghani, Mohammad; Saberi, Amin
作者单位：Stanford University; Stanford University
摘要：This paper makes three contributions to estimating the number of per-fect matching in bipartite graphs. First, we prove that the popular sequential importance sampling algorithm works in polynomial time for dense bipartite graphs. More carefully, our algorithm gives a (1 +/- e)-approximation for the number of perfect matchings of a A.-dense bipartite graph, using O (n 1-2A. A. E-2 ) samples. With size n on each side and for 21 > A. > 0, a A.-dense bipartite graph has all degrees greater than (...
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作者：Bayraktar, Erhan; Chakraborty, Suman; Wu, Ruoyu
作者单位：University of Michigan System; University of Michigan; Uppsala University; Iowa State University
摘要：We consider heterogeneously interacting diffusive particle systems and their large population limit. The interaction is of mean field type with weights characterized by an underlying graphon. A law of large numbers result is established as the system size increases and the underlying graphons converge. The limit is given by a graphon mean field system consisting of independent but heterogeneous nonlinear diffusions whose probability distributions are fully coupled. Well-posedness, continuity a...
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作者：Hermon, Jonathan; Olesker-Taylor, Sam
作者单位：University of British Columbia; University of Warwick
摘要：Consider the random Cayley graph of a finite Abelian group G with re-spect to k generators chosen uniformly at random, with 1 << log k << log |G|. Draw a vertex U <^> Unif(G).We show that the graph distance dist(id, U) from the identity to U con-centrates at a particular value M, which is the minimal radius of a ball in Zk of cardinality at least |G|, under mild conditions. In other words, the distance from the identity for all but o(|G|) of the elements of G lies in the interval [M -o(M), M +...
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作者：Ding, Xiucai; Ji, Hong Chang
作者单位：University of California System; University of California Davis; Institute of Science & Technology - Austria
摘要：Consider the random matrix model A1/2UB U*A1/2, where A and B are two N x N deterministic matrices and U is either an N x N Haar unitary or orthogonal random matrix. It is well known that on the macroscopic scale (Invent. Math. 104 (1991) 201-220), the limiting empirical spectral distribu-tion (ESD) of the above model is given by the free multiplicative convolution of the limiting ESDs of A and B, denoted as & mu;& alpha; & REG; & mu;⠃, where & mu;& alpha; and & mu;⠃ are the limiting ESDs of...
