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作者：Ben Arous, Gerard; Gheissari, Reza; Jagannath, Aukosh
作者单位：New York University; University of Waterloo
摘要：We study the algorithmic thresholds for principal component analysis of Gaussian k-tensors with a planted rank-one spike, via Langevin dynamics and gradient descent. In order to efficiently recover the spike from natural initializations, the signal-to-noise ratio must diverge in the dimension. Our proof shows that the mechanism for the success/failure of recovery is the strength of the curvature of the spike on the maximum entropy region of the initial data. To demonstrate this, we study the d...
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作者：Duminil-Copin, Hugo; Ganguly, Shirshendu; Hammond, Alan; Manolescu, Ioan
作者单位：Universite Paris Saclay; University of California System; University of California Berkeley; University of California System; University of California Berkeley; University of Fribourg
摘要：Let c(n) = c(n)(d) denote the number of self-avoiding walks of length n starting at the origin in the Euclidean nearest-neighbour lattice Z(d). Let mu, = lim(n) c(n)(1/n) denote the connective constant of Z(d). In 1962, Hammersley and Welsh (Quart. J. Math. Oxford Ser (2) 13 (1962) 108-110) proved that, for each d >= 2, there exists a constant C > 0 such that c(n) <= exp(C-n(1/2))mu(n) for all n is an element of N. While it is anticipated that c(n)mu(-n). has a power-law growth in n, the best-...
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作者：Jego, Antoine
作者单位：University of Vienna
摘要：We construct the analogue of Gaussian multiplicative chaos measures for the local times of planar Brownian motion by exponentiating the square root of the local times of small circles. We also consider a flat measure supported on points whose local time is within a constant of the desired thickness level and show a simple relation between the two objects. Our results extend those of (Ann. Probab. 22 (1994) 566-625), and in particular, cover the entire L-1-phase or subcritical regime. These res...
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作者：Aidekon, Elie; Hu, Yueyun; Shi, Zhan
作者单位：Sorbonne Universite; Universite Paris Cite
摘要：It is well known (see Dvoretzky, Erdos and Kakutani (Bull. Res. Council Israel Sect. F 7F (1958) 175-180) and Le Gall (J. Funct. Anal. 71 (1987) 246-262)) that a planar Brownian motion (B-t)(t >= 0) has points of infinite multiplicity, and these points form a dense set on the range. Our main result is the construction of a family of random measures, denoted by {M-infinity(alpha)}(0<2), that are supported by the set of the points of infinite multiplicity. We prove that for any alpha is an eleme...
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作者：Hairer, Martin; Li, Xue-Mei
作者单位：Imperial College London
摘要：We consider slow/fast systems where the slow system is driven by fractional Brownian motion with Hurst parameter H > 1/2. We show that unlike in the case H = 1/2, convergence to the averaged solution takes place in probability and the limiting process solves the 'naively' averaged equation. Our proof strongly relies on the recently obtained stochastic sewing lemma.
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作者：Kosloff, Zemer; Soo, Terry
作者单位：Hebrew University of Jerusalem; University of London; University College London
摘要：Ornstein and Shields (Advances in Math. 10 (1973) 143-146) proved that Brownian motion reflected on a bounded region is an infinite entropy Bernoulli flow, and, thus, Ornstein theory yielded the existence of a measure-preserving isomorphism between any two such Brownian motions. For fixed h > 0, we construct by elementary methods, isomorphisms with almost surely finite coding windows between Brownian motions reflected on the intervals [0, qh] for all positive rationals q.
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作者：Spinka, Yinon
作者单位：University of British Columbia
摘要：We show that any finitely dependent invariant process on a transitive amenable graph is a finitary factor of an i.i.d. process. With an additional assumption on the geometry of the graph, namely that no two balls with different centers are identical, we further show that the i.i.d. process may be taken to have entropy arbitrarily close to that of the finitely dependent process. As an application, we give an affirmative answer to a question of Holroyd (Ann. Inst. Henri Poincare Probab. Stat. 53...
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作者：Benoist, Stephane; Dumaz, Laure; Werner, Wendelin
作者单位：Massachusetts Institute of Technology (MIT); Universite PSL; Universite Paris-Dauphine; Centre National de la Recherche Scientifique (CNRS); Swiss Federal Institutes of Technology Domain; ETH Zurich
摘要：We study random two-dimensional spanning forests in the plane that can be viewed both in the discrete case and in their appropriately taken scaling limits as a uniformly chosen spanning tree with some Poissonian deletion of edges or points. We show how to relate these scaling limits to a stationary distribution of a natural coalescent-type Markov process on a state space of abstract graphs with real-valued edge weights. This Markov process can be interpreted as a renormalization flow. This pro...
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作者：Gwynne, Ewain; Miller, Jason
作者单位：University of Cambridge
摘要：We prove that for any metric, which one can associate with a Liouville quantum gravity (LQG) surface for gamma is an element of (0, 2) satisfying certain natural axioms, its geodesics exhibit the following confluence property. For any fixed point z, a.s. any two gamma-LQG geodesics started from distinct points other than z must merge into each other and subsequently coincide until they reach z. This is analogous to the confluence of geodesics property for the Brownian map proven by Le Gall (Ac...
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作者：Barbu, Viorel; Roeckner, Michael
作者单位：Romanian Academy; University of Bielefeld
摘要：We construct weak solutions to the McKean-Vlasov SDE dX(t) = b(X(t), dL(X(t))/dx (X(t)))dt + sigma(X(t), dL(X(t))/dt (X(t)))dW(t) on R-d for possibly degenerate diffusion matrices sigma with X(0) having a given law, which has a density with respect to Lebesgue measure, dx. Here, L-X(t) denotes the law of X (t). Our approach is to first solve the corresponding nonlinear Fokker-Planck equations and then use the well-known superposition principle to obtain weak solutions of the above SDE.
