Probability Theory and Related Fields最新文献

Let f be a Gaussian random field on (mathbb {R}^d) and let X be the number of critical points of f contained in a compact subset. A long-standing conjecture is that, under mild regularity and non-degeneracy conditions on f, the random variable X has finite moments. So far, this has been established only for moments of order lower than three. In this paper, we prove the conjecture. Precisely, we show that X has finite moment of order p, as soon as, at any given point, the Taylor polynomial of order p of f is non-degenerate. We present a simple and general approach that is not specific to critical points and we provide various applications. In particular, we show the finiteness of moments of the nodal volumes and the number of critical points of a large class of smooth, or holomorphic, Gaussian fields, including the Bargmann-Fock ensemble.

Abstract

Co-evolving network models, wherein dynamics such as random walks on the network influence the evolution of the network structure, which in turn influences the dynamics, are of interest in a range of domains. While much of the literature in this area is currently supported by numerics, providing evidence for fascinating conjectures and phase transitions, proving rigorous results has been quite challenging. We propose a general class of co-evolving tree network models driven by local exploration, started from a single vertex called the root. New vertices attach to the current network via randomly sampling a vertex and then exploring the graph for a random number of steps in the direction of the root, connecting to the terminal vertex. Specific choices of the exploration step distribution lead to the well-studied affine preferential attachment and uniform attachment models, as well as less well understood dynamic network models with global attachment functionals such as PageRank scores (Chebolu and Melsted, in: SODA, 2008). We obtain local weak limits for such networks and use them to derive asymptotics for the limiting empirical degree and PageRank distribution. We also quantify asymptotics for the degree and PageRank of fixed vertices, including the root, and the height of the network. Two distinct regimes are seen to emerge, based on the expected exploration distance of incoming vertices, which we call the ‘fringe’ and ‘non-fringe’ regimes. These regimes are shown to exhibit different qualitative and quantitative properties. In particular, networks in the non-fringe regime undergo ‘condensation’ where the root degree grows at the same rate as the network size. Networks in the fringe regime do not exhibit condensation. Further, non-trivial phase transition phenomena are shown to arise for: (a) height asymptotics in the non-fringe regime, driven by the subtle competition between the condensation at the root and network growth; (b) PageRank distribution in the fringe regime, connecting to the well known power-law hypothesis. In the process, we develop a general set of techniques involving local limits, infinite-dimensional urn models, related multitype branching processes and corresponding Perron–Frobenius theory, branching random walks, and in particular relating tail exponents of various functionals to the scaling exponents of quasi-stationary distributions of associated random walks. These techniques are expected to shed light on a variety of other co-evolving network models.

We find the scaling limits of a general class of boundary-to-boundary connection probabilities and multiple interfaces in the critical planar FK-Ising model, thus verifying predictions from the physics literature. We also discuss conjectural formulas using Coulomb gas integrals for the corresponding quantities in general critical planar random-cluster models with cluster-weight ({q in [1,4)}). Thus far, proofs for convergence, including ours, rely on discrete complex analysis techniques and are beyond reach for other values of q than the FK-Ising model ((q=2)). Given the convergence of interfaces, the conjectural formulas for other values of q could be verified similarly with relatively minor technical work. The limit interfaces are variants of (text {SLE}_kappa ) curves (with (kappa = 16/3) for (q=2)). Their partition functions, that give the connection probabilities, also satisfy properties predicted for correlation functions in conformal field theory (CFT), expected to describe scaling limits of critical random-cluster models. We verify these properties for all (q in [1,4)), thus providing further evidence of the expected CFT description of these models.

We consider general mixed p-spin mean field spin glass models and provide a method to prove that the spectral gap of the Dirichlet form associated with the Gibbs measure is of order one at sufficiently high temperature. Our proof is based on an iteration scheme relating the spectral gap of the N-spin system to that of suitably conditioned subsystems.

In this paper, we study the online nearest neighbor random tree in dimension (din {mathbb {N}}) (called d-NN tree for short) defined as follows. We fix the torus ({mathbb {T}}^d_n) of dimension d and area n and equip it with the metric inherited from the Euclidean metric in ({mathbb {R}}^d). Then, embed consecutively n vertices in ({mathbb {T}}^d_n) uniformly at random and independently, and let each vertex but the first one connect to its (already embedded) nearest neighbor. Call the resulting graph (G_n). We show multiple results concerning the degree sequence of (G_n). First, we prove that typically the number of vertices of degree at least (kin {mathbb {N}}) in the d-NN tree decreases exponentially with k and is tightly concentrated by a new Lipschitz-type concentration inequality that may be of independent interest. Second, we obtain that the maximum degree of (G_n) is of logarithmic order. Third, we give explicit bounds for the number of leaves that are independent of the dimension and also give estimates for the number of paths of length two. Moreover, we show that typically the height of a uniformly chosen vertex in (G_n) is ((1+o(1))log n) and the diameter of ({mathbb {T}}^d_n) is ((2e+o(1))log n), independently of the dimension. Finally, we define a natural infinite analog (G_{infty }) of (G_n) and show that it corresponds to the local limit of the sequence of finite graphs ((G_n)_{n ge 1}). Furthermore, we prove almost surely that (G_{infty }) is locally finite, that the simple random walk on (G_{infty }) is recurrent, and that (G_{infty }) is connected.

The Erdős–Taylor theorem (Acta Math Acad Sci Hungar, 1960) states that if (textsf{L}_N) is the local time at zero, up to time 2N, of a two-dimensional simple, symmetric random walk, then (tfrac{pi }{log N} ,textsf{L}_N) converges in distribution to an exponential random variable with parameter one. This can be equivalently stated in terms of the total collision time of two independent simple random walks on the plane. More precisely, if (textsf{L}_N^{(1,2)}=sum _{n=1}^N mathbb {1}_{{S_n^{(1)}= S_n^{(2)}}}), then (tfrac{pi }{log N}, textsf{L}^{(1,2)}_N) converges in distribution to an exponential random variable of parameter one. We prove that for every (h geqslant 3), the family ( big { frac{pi }{log N} ,textsf{L}_N^{(i,j)} big }_{1leqslant i<jleqslant h}), of logarithmically rescaled, two-body collision local times between h independent simple, symmetric random walks on the plane converges jointly to a vector of independent exponential random variables with parameter one, thus providing a multivariate version of the Erdős–Taylor theorem. We also discuss connections to directed polymers in random environments.

A variety of physical phenomena involve the nonlinear transfer of energy from weakly damped modes subjected to external forcing to other modes which are more heavily damped. In this work we explore this in (finite-dimensional) stochastic differential equations in ({mathbb {R}}^n) with a quadratic, conservative nonlinearity B(x, x) and a linear damping term—Ax which is degenerate in the sense that (textrm{ker} A ne emptyset ). We investigate sufficient conditions to deduce the existence of a stationary measure for the associated Markov semigroups. Existence of such measures is straightforward if A is full rank, but otherwise, energy could potentially accumulate in (textrm{ker} A) and lead to almost-surely unbounded trajectories, making the existence of stationary measures impossible. We give a relatively simple and general sufficient condition based on time-averaged coercivity estimates along trajectories in neighborhoods of (textrm{ker} A) and many examples where such estimates can be made.

We study reflected diffusion on uniform domains where the underlying space admits a symmetric diffusion that satisfies sub-Gaussian heat kernel estimates. A celebrated theorem of Jones (Acta Math 147(1-2):71–88, 1981) states that uniform domains in Euclidean space are extension domains for Sobolev spaces. In this work, we obtain a similar extension property for metric spaces equipped with a Dirichlet form whose heat kernel satisfies a sub-Gaussian estimate. We introduce a scale-invariant version of this extension property and apply it to show that the reflected diffusion process on such a uniform domain inherits various properties from the ambient space, such as Harnack inequalities, cutoff energy inequality, and sub-Gaussian heat kernel bounds. In particular, our work extends Neumann heat kernel estimates of Gyrya and Saloff-Coste (Astérisque 336:145, 2011) beyond the Gaussian space-time scaling. Furthermore, our estimates on the extension operator imply that the energy measure of the boundary of a uniform domain is always zero. This property of the energy measure is a broad generalization of Hino’s result (Probab Theory Relat Fields 156:739–793, 2013) that proves the vanishing of the energy measure on the outer square boundary of the standard Sierpiński carpet equipped with the self-similar Dirichlet form.

Abstract

We consider the following stochastic heat equation $$begin{aligned} partial _t u(t,x) = tfrac{1}{2} partial ^2_x u(t,x) + b(u(t,x)) + sigma (u(t,x)) {dot{W}}(t,x), end{aligned}$$ defined for ((t,x)in (0,infty )times {mathbb {R}}) , where ({dot{W}}) denotes space-time white noise. The function (sigma ) is assumed to be positive, bounded, globally Lipschitz, and bounded uniformly away from the origin, and the function b is assumed to be positive, locally Lipschitz and nondecreasing. We prove that the Osgood condition $$begin{aligned} int _1^infty frac{textrm{d}y}{b(y)}<infty end{aligned}$$ implies that the solution almost surely blows up everywhere and instantaneously, In other words, the Osgood condition ensures that (textrm{P}{ u(t,x)=infty quad hbox { for all } t>0 hbox { and } xin {mathbb {R}}}=1.) The main ingredients of the proof involve a hitting-time bound for a class of differential inequalities (Remark 3.3), and the study of the spatial growth of stochastic convolutions using techniques from the Malliavin calculus and the Poincaré inequalities that were developed in Chen et al. (Electron J Probab 26:1–37, 2021, J Funct Anal 282(2):109290, 2022).

We investigate the quadratic Schrödinger bridge problem, a.k.a. Entropic Optimal Transport problem, and obtain weak semiconvexity and semiconcavity bounds on Schrödinger potentials under mild assumptions on the marginals that are substantially weaker than log-concavity. We deduce from these estimates that Schrödinger bridges satisfy a logarithmic Sobolev inequality on the product space. Our proof strategy is based on a second order analysis of coupling by reflection on the characteristics of the Hamilton–Jacobi–Bellman equation that reveals the existence of new classes of invariant functions for the corresponding flow.