We recently introduced a robust approach to the derivation of sharp asymptotic formula for correlation functions of statistical mechanics models in the high-temperature regime. We describe its application to the nonperturbative proof of Ornstein-Zernike asymptotics of 2-point functions for self-avoiding walks, Bernoulli percolation and ferromagnetic Ising models. We then extend the proof, in the Ising case, to arbitrary odd-odd correlation functions. We discuss the fluctuations of connection paths (invariance principle), and relate the variance of the limiting process to the geometry of the equidecay profiles. Finally, we explain the relation between these results from Statistical Mechanics and their counterparts in Quantum Field Theory.
For a family of random-cluster models with cluster weights $q\geq 1$, we prove that the probability that $0$ is connected to $x$ is asymptotically equal to $\tfrac{1}{q}χ(β)^{2}βJ_{0,x}$. The method developed in this article can be applied to any spin model for which there exists a random-cluster representation which is one-monotonic.
We prove that the correction to exponential decay of the truncated two points function in the homogeneous positive field Ising model is $c\|x\|^{-(d-1)/2}$. The proof is based on the development in the random current representation of a "modern" Ornstein-Zernike theory, as developed by Campanino, Ioffe and Velenik.
We provide O(1) asymptotics for the average number of deep minima of the (p,k) spiked tensor model. We also derive an explicit formula for the limiting ground state energy on the N-dimensional sphere, similar to the work of Jagannath-Lopatto-Miolane. Moreover, when the signal to noise ratio is large enough, the expected number of deep minima is asymptotically finite as N tends to infinity and we determine its limit as the signal-to-noise ratio diverges.
In this paper we study the metastable behavior of one of the simplest disordered spin system, the random field Curie-Weiss model. We will show how the potential theoretic approach can be used to prove sharp estimates on capacities and metastable exit times also in the case when the distribution of the random field is continuous. Previous work was restricted to the case when the random field takes only finitely many values, which allowed the reduction to a finite dimensional problem using lumping techniques. Here we produce the first genuine sharp estimates in a context where entropy is importa…
We offer in this paper the non-asymptotical bilateral sharp exponential estimates for tail of maximum distribution of {\it discontinuous} random fields. Our consideration based on the theory of Prokhorov-Skorokhod spaces of random fields and on the theory of multivariate Banach spaces of random variables with exponential decreasing tails of distributions.
In this work, we consider critical planar site percolation on the triangular lattice and derive sharp estimates on the asymptotics of the probability of half-plane $j$-arm events for $j \geq 1$ and planar (polychromatic) $j$-arm events for $j>1$. These estimates greatly improve previous results and in particular answer (a large part of) a question of Schramm (ICM Proc., 2006).
The sharp asymptotics for the L^2-quantization errors of Gaussian measures on a Hilbert space and, in particular, for Gaussian processes is derived. The condition imposed is regular variation of the eigenvalues.
The current research of fractional Sturm-Liouville boundary value problems focuses on the qualitative theory and numerical methods, and much progress has been recently achieved in both directions. The objective of this paper is to explore a different route, namely, construction of explicit asymptotic approximations for the solutions. As a study case, we consider a problem with left and right Riemann-Liouville derivatives, for which our analysis yields asymptotically sharp estimates for the sequence of eigenvalues and eigenfunctions.
We give sharp estimate for the free energy of directed polymers in random environment in dimension 1+1. This estimate was known for a Gaussian environment, we extend it to the case where the law of the environment is infinitely divisible.
We consider continuous and discrete (1+1)-dimensional wetting models which undergo a localization/delocalization phase transition. Using a simple approach based on Renewal Theory we determine the precise asymptotic behavior of the partition function, from which we obtain the scaling limits of the models and an explicit construction of the infinite volume measure (thermodynamic limit) in all regimes, including the critical one.
The entropy, the spectral radius and the drift are important numerical quantities associated to random walks on countable groups. We prove sharp inequalities relating those quantities for walks with a finite second moment, improving upon previous results of Avez, Varopoulos, Carne, Ledrappier. We also deduce inequalities between these quantities and the volume growth of the group. Finally, we show that the equality case in our inequality is rather rigid.
The effect of disorder for pinning models is a subject which has attracted much attention in theoretical physics and rigorous mathematical physics. A peculiar point of interest is the question of coincidence of the quenched and annealed critical point for a small amount of disorder. The question has been mathematically settled in most cases in the last few years, giving in particular a rigorous validation of the Harris Criterion on disorder relevance. However, the marginal case, where the return probability exponent is equal to $1/2$, i.e. where the inter-arrival law of the renewal process is …
In the present paper, we investigate the collapsed phase of the interacting partially-directed self-avoiding walk (IPDSAW) that was introduced in Zwanzig and Lauritzen (1968). We provide sharp asymptotics of the partition function inside the collapsed phase, proving rigorously a conjecture formulated in Guttmann (2015) and Owczarek et al. (1993). As a by-product of our result, we obtain that, inside the collapsed phase, a typical IPDSAW trajectory is made of a unique macroscopic bead, consisting of a concatenation of long vertical stretches of alternating signs, outside which only finitely man…
We consider a class of parabolic semi-linear stochastic partial differential equations driven by space-time white noise on a compact space interval. Our aim is to obtain precise asymptotics of the transition times between metastable states. A version of the so-called Eyring-Kramers Formula is proven in an infinite dimensional setting. The proof is based on a spatial finite difference discretization of the stochastic partial differential equation. The expected transition time is computed for the finite dimensional approximation and controlled uniformly in the dimension.
The Wasserstein distance between two probability measures on a metric space is a measure of closeness with applications in statistics, probability, and machine learning. In this work, we consider the fundamental question of how quickly the empirical measure obtained from $n$ independent samples from $μ$ approaches $μ$ in the Wasserstein distance of any order. We prove sharp asymptotic and finite-sample results for this rate of convergence for general measures on general compact metric spaces. Our finite-sample results show the existence of multi-scale behavior, where measures can exhibit radic…
The purpose of this paper is to provide a sharp analysis on the asymptotic behavior of the Durbin-Watson statistic. We focus our attention on the first-order autoregressive process where the driven noise is also given by a first-order autoregressive process. We establish the almost sure convergence and the asymptotic normality for both the least squares estimator of the unknown parameter of the autoregressive process as well as for the serial correlation estimator associated to the driven noise. In addition, the almost sure rates of convergence of our estimates are also provided. It allows us …
For an ergodic Brownian diffusion with invariant measure $ν$, we consider a sequence of empirical distributions ($ν$n) n$\ge$1 associated with an approximation scheme with decreasing time step ($γ$n) n$\ge$1 along an adapted regular enough class of test functions f such that f --$ν$(f) is a coboundary of the infinitesimal generator A. Denote by $σ$ the diffusion coefficient and $Φ$ the solution of the Poisson equation A$Φ$ = f -- $ν$(f). When the square norm of |$σ$ * $Φ$| 2 lies in the same coboundary class as f , we establish sharp non-asymptotic concentration bounds for suitable normalizati…
Sharp large deviation results of Bahadur-Ranga Rao type are provided for the $q$-norm of random vectors distributed on the $\ell _{p}^{n}$-ball ${\mathbb{B}}^{n}_{p}$ according to the cone probability measure or the uniform distribution for $1 \le q<p < \infty $, thereby furthering previous large deviation results by Kabluchko, Prochno and Thäle in the same setting. These results are then applied to deduce sharp asymptotics for intersection volumes of different $\ell _{p}^{n}$-balls in the spirit of Schechtman and Schmuckenschläger, and for the length of the projection of an $\ell _{p}^{n}$-ba…
We consider instances of long-range percolation on $\mathbb Z^d$ and $\mathbb R^d$, where points at distance $r$ get connected by an edge with probability proportional to $r^{-s}$, for $s\in (d,2d)$, and study the asymptotic of the graph-theoretical (a.k.a. chemical) distance $D(x,y)$ between $x$ and $y$ in the limit as $|x-y|\to\infty$. For the model on $\mathbb Z^d$ we show that, in probability as $|x|\to\infty$, the distance $D(0,x)$ is squeezed between two positive multiples of $(\log r)^Δ$, where $Δ:=1/\log_2(1/γ)$ for $γ:=s/(2d)$. For the model on $\mathbb R^d$ we show that $D(0,xr)$ is,…
Freidlin-Wentzell theory of large deviations can be used to compute the likelihood of extreme or rare events in stochastic dynamical systems via the solution of an optimization problem. The approach gives exponential estimates that often need to be refined via calculation of a prefactor. Here it is shown how to perform these computations in practice. Specifically, sharp asymptotic estimates are derived for expectations, probabilities, and mean first passage times in a form that is geared towards numerical purposes: they require solving well-posed matrix Riccati equations involving the minimize…
We study boundary non-crossing probabilities $$ P_{f,u} := \mathrm P\big(\forall t\in \mathbb T\ X_t + f(t)\le u(t)\big) $$ for continuous centered Gaussian process $X$ indexed by some arbitrary compact separable metric space $\mathbb T$. We obtain both upper and lower bounds for $P_{f,u}$. The bounds are matching in the sense that they lead to precise logarithmic asymptotics for the large-drift case $P_{y f,u}$, $y \to+\infty$, which are two-term approximations (up to $o(y)$). The asymptotics are formulated in terms of the solution $\tilde f$ to the constrained optimization problem $$ \|h\|_{…
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Duke University（杜克大学）
The non-asymptotic tail bounds of random variables play crucial roles in probability, statistics, and machine learning. Despite much success in developing upper bounds on tail probability in literature, the lower bounds on tail probabilities are relatively fewer. In this paper, we introduce systematic and user-friendly schemes for developing non-asymptotic lower bounds of tail probabilities. In addition, we develop sharp lower tail bounds for the sum of independent sub-Gaussian and sub-exponential random variables, which match the classic Hoeffding-type and Bernstein-type concentration inequal…
This paper is concerned with two related types of directed polymers in a random medium. The first one is a d-dimensional Brownian motion living in a random environment which is Brownian in time and homogeneous in space. The second is a continuous-time random walk on the lattice Z^d, in a random environment with similar properties as in continuous space. The case of a space-time white noise environment can be acheived in this second setting. By means of some Gaussian tools, we estimate the free energy of these models at low temperature, and give some further information on the strong disorder r…
Many real-world networks exhibit the so-called small-world phenomenon: their typical distances are much smaller than their sizes. One mathematical model for this phenomenon is a long-range percolation graph on a $d$-dimensional box $\{0, 1, \cdots, N\}^d$, in which edges are independently added between far-away sites with probability falling off as a power of the Euclidean distance. A natural question is how the resulting diameter of the box of size $N$, measured in graph-theoretical distance, scales with $N$. This question has been intensely studied in the past and the answer depends on the e…
A subset A of {0,1,...,n} is said to be a 2-additive basis for {1,2,...,n} if each j in {1,2,...,n} can be written as j=x+y, x,y in A, x<=y. If we pick each integer in {0,1,...,n} independently with probability p=p_n tending to 0, thus getting a random set A, what is the probability that we have obtained a 2-additive basis? We address this question when the target sum-set is [(1-alpha)n,(1+alpha)n] (or equivalently [alpha n, (2-alpha) n]) for some 0<alpha<1. Under either model, the Stein-Chen method of Poisson approximation is used, in conjunction with Janson's inequalities, to tease out a ver…
We study stochastic volatility models in which the volatility process is a positive continuous function of a continuous Volterra stochastic process. We state some pathwise large deviation principles for the scaled log-price.
Range convergence and sharp one-arm asymptotics for the critical percolation cluster in high dimensions
Facultad de Matemáticas, Pontificia Universidad Católica de Chile（智利天主教Pontificia大学数学学院）；Research Institute for Mathematical Sciences, Kyoto University（京都大学数学科学研究所）；Département de mathématiques et de statistique, Université de Montréal（蒙特利尔大学数学与统计系）；Department of Mathematics, Graduate School of Science, Kyoto University（京都大学理学研究科数学系）
We consider the Moran model on the sharp peak landscape, in the asymptotic regime studied in [3], where a quasispecies is formed. We find explicitly the distribution of this quasispecies.
In this paper, we obtain a large and moderate deviation principle for the law of the maximum of a random Dyck path. Our result extend the results of Chung, Kennedy, Kaigh and Khorunzhiy and Marckert.
We prove that typical (in the model-free finance setting) price paths with jumps may be uniformly approximated with accuracy $c>0$ by paths whose total variation is of order $1/c.$ A more precise result is obtained for semimartingales with jumps.
