論文

Consider the first passage percolation on the d-dimensional lattice Zd with identical and independent weight distributions and the first passage time T. In this paper, we study the upper tail large deviations P(T(0, nx) > n(μ + ξ)), for ξ > 0 and x ≠ 0 with a time constant μ, for weights that satisfy a tail assumption P(τe > t) ≈ β exp (−αtr ). When r ≤ 1 (this includes the well-known Eden growth model), we show that the upper tail large deviation decays as exp (−(2dαξr + o(1))n). When 1 < r ≤ d, we find that the rate function can be naturally described by a variational formula, called the discrete p-Capacity, and we study its asymptotics. The case r = d is critical and logarithmic corrections appear. For r ∈ (1, d), we show that the large deviation event {T(0, nx) > n(μ + ξ)} is described by a localization of high weights around the endpoints. The picture changes for r ≥ d where the configuration is not anymore localized.

We are interested in the asymptotics of the p-capacity between the origin and the set nB, where B is the boundary of the unit ball of the lattice Zd. The p-capacity is defined as the minimum of the Dirichlet energy associated with a discrete version of the p-Laplacian. This variational problem has arisen in particular in the study of large deviations for first passage percolation. For p < d, the p-capacity converges to some positive constant, while for p > d the capacity vanishes polynomially fast. The present paper deals with the case p = d, for which we prove that the p-capacity vanishes as cd(log n)−d+1 with an explicit constant cd. Our proof relies on Thomson’s principle for the p-capacity.

The non-random fluctuation is one of the central objects in first passage percolation. It was proved in [11] that for a particular asymptotic direction, it diverges in a lattice first passage percolation with an explicit lower bound. In this paper, we discuss the non-random fluctuation in Euclidean first passage percolations and show that it diverges in dimension d ≥ 2 in this model also. Compared with the result in [11], the present result is proved for any direction and improves the lower bound.

We study the fluctuations of discretized versions of the stochastic heat equation (SHE) and the Kardar– Parisi–Zhang (KPZ) equation in spatial dimensions d ≥ 3 in the weak disorder regime. The discretization is defined using the directed polymer model. Previous research has identified the scaling limit of both equations under a suboptimal moment condition and, in particular, it was established that both converge in law to the same limit. We extend this result by showing that the fluctuations of both equations are close in probability in the subcritical weak disorder regime, indicating that they share the same scaling limit (the existence of which remains open). Our result applies under a moment condition that is expected to hold throughout the interior of the weak disorder phase, which is currently only known under a technical assumption on the environment. We also prove a lower tail concentration of the partition functions.

The Kardar-Parisi-Zhang equation (KPZ equation) is solved via Cole-Hopf transformation h = log u, where u is the solution of the multiplicative stochastic heat equation(SHE). In [CD20, CSZ20, G20], they consider the solution of two dimensional KPZ equation via the solution u ε of SHE with the flat initial condition and with noise which is mollified in space on scale in " and its strength is weakened as (Formula Presented), and they prove that when (Formula Presented) converges in distribution as a random field to a solution of Edwards-Wilkinson equation. In this paper, we consider a stochastic heat equation u" with a general initial condition u0 and its transformation F(u") for F in a class of functions F, which contains F(x) = xp (0 < p ≤ 1) and F(x) = log x. Then, we prove that 1 (Formula Presented) converges in distribution as a random field to a centered Gaussian field jointly in finitely many F 2 F, t, and u0. In particular, we show the fluctuations of solutions of stochastic heat equations and KPZ equations jointly converge to solutions of SPDEs which depend on u0. Our main tools are Itô’s formula, the martingale central limit theorem, and the homogenization argument as in [CNN22]. To this end, we also prove a local limit theorem for the partition function of intermediate disorder 2d directed polymers.

We consider the Ising perceptron model with N spins and M “Nα patterns, with a general activation function U that is bounded above. For U bounded away from zero or Upxq “1tx ě κu, it was shown by Talagrand (2000, 2011b) that for small densities α, the free energy of the model converges as N Ñ 8 to the replica symmetric formula conjectured in the physics literature by Krauth and Mézard (1989) (see also Gardner and Derrida, 1988). We give a new proof of this result, which covers the more general class of all functions U that are bounded above and satisfy a certain variance bound. The proof uses the (first and second) moment method conditional on the approximate message passing iterates of the model. In order to deduce our main theorem, we also prove a new concentration result for the perceptron model in the case where U is not bounded away from zero.

We consider the discrete directed polymer model with i.i.d. environment and we study the fluctuations of the tail n(d-2)/4(W∞-Wn) of the normalized partition function. It was proven by Comets and Liu (J. Math. Anal. Appl. 455 (2017) 312-335), that for sufficiently high temperature, the fluctuations converge in distribution towards the product of the limiting partition function and an independent Gaussian random variable. We extend the result to the whole L2-region of temperature, which is predicted to be the maximal high-temperature region where the Gaussian fluctuations should occur under the considered scaling. To do so, we manage to avoid the heavy 4th-moment computation and instead rely on the local limit theorem for polymers (Fund. Math. 147 (1995) 173-180; Ann. Inst. Henri Poincaré Probab. Stat. 42 (2006) 521-534) and homogenization.

When can the input of a ReLU neural network be inferred from its output? In other words, when is the network injective? We consider a single layer, x↦ReLU(Wx), with a random Gaussian m×n matrix W, in a high-dimensional setting where n,m→∞. Recent work connects this problem to spherical integral geometry giving rise to a conjectured sharp injectivity threshold for α=m/n by studying the expected Euler characteristic of a certain random set. We adopt a different perspective and show that injectivity is equivalent to a property of the ground state of the spherical perceptron, an important spin glass model in statistical physics. By leveraging the (non-rigorous) replica symmetry-breaking theory, we derive analytical equations for the threshold whose solution is at odds with that from the Euler characteristic. Furthermore, we use Gordon's min–max theorem to prove that a replica-symmetric upper bound refutes the Euler characteristic prediction. Along the way we aim to give a tutorial-style introduction to key ideas from statistical physics in an effort to make the exposition accessible to a broad audience. Our analysis establishes a connection between spin glasses and integral geometry but leaves open the problem of explaining the discrepancies.

There have been recently several works studying the regularized stochastic heat equation (SHE) and Kardar–Parisi–Zhang (KPZ) equation in dimension d≥3 as the smoothing parameter is switched off, but most of the results did not hold in the full temperature regions where they should. Inspired by martingale techniques coming from the directed polymers literature, we first extend the law of large numbers for SHE obtained in Mukherjee et al. (2016) to the full weak disorder region of the associated polymer model and to more general initial conditions. We further extend the Edwards–Wilkinson regime of the SHE and KPZ equation studied in Gu et al. (2018), Magnen and Unterberger (2018), Dunlap et al. (2020) to the full L2-region, along with multidimensional convergence and general initial conditions for the KPZ equation (and SHE), which were not proven before. To do so, we rely on a martingale CLT combined with a refinement of the local limit theorem for polymers.

In this article, we consider a generalized First-passage percolation model, where each edge in Zd is independently assigned an infinite weight with probability 1−p, and a random finite weight otherwise. The existence and positivity of the time constant have been established in Cerf and Théret (2016). Recently, using sophisticated multi-scale renormalizations, Cerf and Dembin (2022) proved that the time constant of chemical distance in super-critical percolation is Lipschitz continuous. In this work, we propose a different approach leveraging lattice animal theory and a simple one-step renormalization with the aid of Russo's formula, to show the Lipschitz continuity of the time constant in generalized First-passage percolation.

We consider the frog model with Bernoulli initial configuration, which is an interacting particle system on the multidimensional lattice consisting of two states of particles: active and sleeping. Active particles perform independent simple random walks. On the other hand, although sleeping particles do not move at first, they become active and can move around when touched by active particles. Initially, only the origin has one active particle, and the other sites have sleeping particles according to a Bernoulli distribution. Then, starting from the original active particle, active ones are gradually generated and propagate across the lattice, with time. It is of interest to know how the propagation of active particles behaves as the parameter of the Bernoulli distribution varies. In this paper, we treat the so-called time constant describing the speed of propagation, and prove that the absolute difference between the time constants for parameters p,q∈(0,1] is bounded from above and below by multiples of |p-q|.

In this paper, we study the maximal edge-traversal time on optimal paths in First-passage percolation on the lattice Zd for several edge distributions, including the Pareto and Weibull distributions. It is known to be unbounded when the edge distribution has unbounded support [J. van den Berg and H. Kesten. Inequalities for the time constant in first-passage percolation. Ann. Appl. Probab. 56-80, 1993]. We determine the order of the growth depending on the tail of the edge distribution.

We study a family of Ising perceptron models with {0, 1}-valued activation functions. This includes the classical half-space models, as well as some of the symmetric models considered in recent works. For each of these models we show that the free energy is self-averaging, there is a sharp threshold sequence, and the free energy is universal with respect to the disorder. A prior work of C. Xu (2019) used very different methods to show a sharp threshold sequence in the half-space Ising perceptron with Bernoulli disorder. Recent works of Perkins-Xu (2021) and Abbe-Li-Sly (2021) determined the sharp threshold and limiting free energy in a symmetric perceptron model. The results of this paper apply in more general settings, and are based on new “add one constraint” estimates extending Talagrand's estimates for the half-space model (1999, 2011).

We study isotropic Gaussian random fields on the high-dimensional sphere with an added deterministic linear term, also known as mixed p-spin Hamiltonians with external field. We prove that if the external field is sufficiently strong, then the resulting function has trivial geometry, that is only two critical points. This contrasts with the situation of no or weak external field where these functions typically have an exponential number of critical points. We give an explicit threshold hc for the magnitude of the external field necessary for trivialization and conjecture hc to be sharp. The Kac–Rice formula is our main tool. Our work extends Fyodorov [14], which identified the trivial regime for the special case of pure p-spin Hamiltonians with random external field.

In this paper, we study the upper tail large deviation for the one-dimensional frog model. In this model, sleeping and active frogs are assigned to sites of Z. While sleeping frogs do not move, the active ones move as independent simple random walks and activate any sleeping frogs. An interest of this model is the asymptotic behavior of the first passage time T(0,n), which is the time needed to activate the sleeping frog at the site n, assuming there is only one active frog at 0 at the beginning. While the law of large numbers and central limit theorems have been well established, the intricacies of large deviations remain elusive. Using renewal theory, Bérard, J., Ramírez, A.F. (Ann. Probab. 44(4), 2770–2816, 2016) have pointed out a slowdown phenomenon where the probability that the first passage time T(0,n) is significantly larger than its expectation decays sub-exponentially and lies between exp(-n1/2+o(1)) and exp(-n1/3+o(1)). In this article, using a novel covering process approach, we confirm that 1/2 is the correct exponent, i.e., the rate of upper large deviations is given by n1/2. Moreover, we obtain an explicit rate function that is characterized by properties of Brownian motion and is strictly concave.