Probability Theory and Related Fields最新文献

We construct an additive functional playing the role of the local time—at a fixed point x—for Markov processes indexed by Lévy trees. We start by proving that Markov processes indexed by Lévy trees satisfy a special Markov property which can be thought as a spatial version of the classical Markov property. Then, we construct our additive functional by an approximation procedure and we characterize the support of its Lebesgue-Stieltjes measure. We also give an equivalent construction in terms of a special family of exit local times. Finally, combining these results, we show that the points at which the Markov process takes the value x encode a new Lévy tree and we construct explicitly its height process. In particular, we recover a recent result of Le Gall concerning the subordinate tree of the Brownian tree where the subordination function is given by the past maximum process of Brownian motion indexed by the Brownian tree.

We introduce a natural measure on bi-infinite random walk trajectories evolving in a time-dependent environment driven by the Langevin dynamics associated to a gradient Gibbs measure with convex potential. We derive an identity relating the occupation times of the Poissonian cloud induced by this measure to the square of the corresponding gradient field, which—generically—is not Gaussian. In the quadratic case, we recover a well-known generalization of the second Ray–Knight theorem. We further determine the scaling limits of the various objects involved in dimension 3, which are seen to exhibit homogenization. In particular, we prove that the renormalized square of the gradient field converges under appropriate rescaling to the Wick-ordered square of a Gaussian free field on (mathbb R^3) with suitable diffusion matrix, thus extending a celebrated result of Naddaf and Spencer regarding the scaling limit of the field itself.

It is a classical observation that lacunary function systems exhibit many properties which are typical for systems of independent random variables. However, it had already been observed by Erdős and Fortet in the 1950s that probability theory’s limit theorems may fail for lacunary sums (sum f(n_k x)) if the sequence ((n_k)_{k ge 1}) has a strong arithmetic “structure”. The presence of such structure can be assessed in terms of the number of solutions (k,ell ) of two-term linear Diophantine equations (a n_k - b n_ell = c). As the first author proved with Berkes in 2010, saving an (arbitrarily small) unbounded factor for the number of solutions of such equations compared to the trivial upper bound, rules out pathological situations as in the Erdős–Fortet example, and guarantees that (sum f(n_k x)) satisfies the central limit theorem (CLT) in a form which is in accordance with true independence. In contrast, as shown by the first author, for the law of the iterated logarithm (LIL) the Diophantine condition which suffices to ensure “truly independent” behavior requires saving this factor of logarithmic order. In the present paper we show that, rather surprisingly, saving such a logarithmic factor is actually the optimal condition in the LIL case. This result reveals the remarkable fact that the arithmetic condition required of ((n_k)_{k ge 1}) to ensure that (sum f(n_k x)) shows “truly random” behavior is a different one at the level of the CLT than it is at the level of the LIL: the LIL requires a stronger arithmetic condition than the CLT does.

We construct and study properties of an infinite dimensional analog of Kahane’s theory of Gaussian multiplicative chaos (Kahane in Ann Sci Math Quebec 9(2):105-150, 1985). Namely, if (H_T(omega )) is a random field defined w.r.t. space-time white noise (dot{B}) and integrated w.r.t. Brownian paths in (dge 3), we consider the renormalized exponential (mu _{gamma ,T}), weighted w.r.t. the Wiener measure (mathbb {P}_0(textrm{d}omega )). We construct the almost sure limit (mu _gamma = lim _{Trightarrow infty } mu _{gamma ,T}) in the entire weak disorder (subcritical) regime and call it subcritical GMC on the Wiener space. We show that

meaning that (mu _gamma ) is supported almost surely only on (gamma )-thick paths, and consequently, the normalized version is singular w.r.t. the Wiener measure. We then characterize uniquely the limit (mu _gamma ) w.r.t. the mollification scheme (phi ) in the sense of Shamov (J Funct Anal 270:3224–3261, 2016) – we show that the law of (dot{B}) under the random rooted measure (mathbb Q_{mu _gamma }(textrm{d}dot{B}textrm{d}omega )= mu _gamma (textrm{d}omega ,dot{B})P(textrm{d}dot{B})) is the same as the law of the distribution (fmapsto dot{B}(f)+ gamma int _0^infty int _{mathbb {R}^d} f(s,y) phi (omega _s-y) textrm{d}s textrm{d}y) under (P otimes mathbb {P}_0). We then determine the fractal properties of the measure around (gamma )-thick paths: (-C_2 le liminf _{varepsilon downarrow 0} varepsilon ^2 log {widehat{mu }}_gamma (Vert omega Vert< varepsilon ) le limsup _{varepsilon downarrow 0}sup _eta varepsilon ^2 log {widehat{mu }}_gamma (Vert omega -eta Vert < varepsilon ) le -C_1) w.r.t a weighted norm (Vert cdot Vert ). Here (C_1>0) and (C_2<infty ) are the uniform upper (resp. pointwise lower) Hölder exponents which are explicit in the entire weak disorder regime. Moreover, they converge to the scaling exponent of the Wiener measure as the disorder approaches zero. Finally, we establish negative and (L^p) ((p>1)) moments for the total mass of (mu _gamma ) in the weak disorder regime.

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.