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作者：Hao, Zimo; Zhang, Xicheng; Zhu, Rongchan; Zhu, Xiangchan
作者单位：University of Bielefeld; Beijing Institute of Technology; Chinese Academy of Sciences; Academy of Mathematics & System Sciences, CAS
摘要：In this paper we study singular kinetic equations on R-2d by the paracontrolled distribution method introduced in Gubinelli, Imkeller and Perkowski (Forum Math. Pi 3 (2015) e6-75). We first develop paracontrolled calculus in the kinetic setting and use it to establish the global well-posedness for the linear singular kinetic equations under the assumptions that the products of singular terms are well defined. We also demonstrate how the required products can be defined in the case that singula...
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作者：Minzer, Dor; Sah, Ashwin; Sawhney, Mehtaab
作者单位：Massachusetts Institute of Technology (MIT)
摘要：We prove that there exists a constant gamma (crit) approximate to 0.17566 such that if G similar to G (n, 1/2), then for any epsilon > 0 with high probability G has a equipartition such that each vertex has (gamma( crit) - epsilon) root n more neighbors in its own part than in the other part and with high probability no such partition exists for a separation of (gamma( crit) + epsilon) root n . The proof involves a number of tools ranging from isoperimetric results on vertex-transitive sets of...
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作者：Hansen, Benjamin; Mueller, Tobias
作者单位：University of Groningen
摘要：We consider percolation on the Voronoi tessellation generated by a homogeneous Poisson point process on the hyperbolic plane. We show that the critical probability for the existence of an infinite cluster is asymptotically equal to pi A/3 as A-* 0. This answers a question of Benjamini and Schramm ( J. Amer. Math. Soc. 14 (2001) 487-507).
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作者：Dareiotis, Konstantinos; Gerencser, Mate
作者单位：University of Leeds; Technische Universitat Wien
摘要：Differential equations perturbed by multiplicative fractional Brownian motions are considered. Depending on the value of the Hurst parameter H, the resulting equation is pathwise viewed as an ODE, YDE, or RDE. In all three regimes, we show regularisation by noise phenomena by proving the strongest kind of well-posedness with irregular drift: strong existence and path-by-path uniqueness. In the Young and smooth regime H > 1/2, the condition on the drift coefficient is optimal in the sense that ...
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作者：Harris, Simon; Johnston, Samuel g. G.; Pardo, Juan carlos
作者单位：University of Auckland; University of London; King's College London
摘要：Consider a population evolving as a critical continuous-time Galton- Watson (GW) tree. Conditional on the population surviving until a large time T, sample k individuals uniformly at random (without replacement) from amongst those alive at time T. What is the genealogy of this sample of individuals? In cases where the offspring distribution has finite variance, the probabilistic properties of the joint ancestry of these k particles are well J. Probab. 24 (2019) 1-35). In the present article, w...
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作者：Adhikari, Arka; Chatterjee, Sourav
作者单位：Stanford University; Stanford University
摘要：Consider a discrete one-dimensional random surface whose height at a point grows as a function of the heights at neighboring points, plus an independent random noise. Assuming that this function is equivariant under constant shifts, symmetric in its arguments, and at least six times continuously differentiable in a neighborhood of the origin, we show that, as the variance of the noise goes to zero, any such process converges to the Cole-Hopf solution of the 1D KPZ equation under a suitable sca...
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作者：Bethencourt, Loic
作者单位：Centre National de la Recherche Scientifique (CNRS); Universite Paris Cite; Sorbonne Universite
摘要：We consider a particle with position (X-t)(t >= 0) living in R+, whose velocity (V-t)(t >= 0) is a positive recurrent diffusion with heavy-tailed invariant distribution when the particle lives in (0, infinity). When it hits the boundary x = 0, the particle restarts with a random strictly positive velocity. We show that the properly rescaled position process converges weakly to a stable process reflected on its infimum. From a P.D.E. point of view, the time-marginals of (X-t, V-t)(t >= 0) solve...
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作者：Gorin, Vadim; Huang, Jiaoyang
作者单位：University of California System; University of California Berkeley; University of California System; University of California Berkeley; University of Pennsylvania
摘要：We introduce dynamical versions of loop (or Dyson-Schwinger) equations for large families of two-dimensional interacting particle systems, including Dyson Brownian motion, Nonintersecting Bernoulli/Poisson random walks, beta -corners processes, uniform and Jack-deformed measures on Gelfand-Tsetlin patterns, Macdonald processes, and (q, kappa)-distributions on lozenge tilings. Under technical assumptions we show that the dynamical loop equations lead to Gaussian field type fluctuations. As an a...
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作者：Basu, Riddhipratim; Bhatia, Manan; Ganguly, Shirshendu
作者单位：Tata Institute of Fundamental Research (TIFR); University of California System; University of California Berkeley
摘要：First passage percolation (FPP) on Z d or R d is a canonical model of a random metric space where the standard Euclidean geometry is distorted by random noise. Of central interest is the length and the geometry of the geodesic, the shortest path between points. Since the latter, owing to its length minimization, traverses through atypically low values of the underlying noise variables, it is an important problem to quantify the disparity between the environment rooted at a point on the geodesi...
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作者：Narayanan, Hariharan; Sheffield, Scott
作者单位：Tata Institute of Fundamental Research (TIFR); Massachusetts Institute of Technology (MIT)
摘要：Suppose a, beta are Lipschitz, strongly concave functions from [0, 1] to R and y is a concave function from [0, 1] to R such that a(0) = y(0) = 0, a(1)=8(0) = 0 and 8(1)=y(1) = 0. For an nxn Hermitian matrix W, let spec(W) denote the vector in R whose coordinates are the eigenvalues of W listed in nonincreasing order. Let lambda=aa, mu=-beta on (0, 1) and v=ay, at all points of (0, 1], where a is the left derivative. Let lambda(i);= n2(a)-a()), for i = [n], and similarly, () == n2(B()-B(2)) an...
