相关论文: Generalised directed last passage percolation: inv…

Directed last passage percolation models on the plane, where one studies the weight as well as the geometry of optimizing paths (called polymers) in a field of i.i.d. weights, are paradigm examples of models in the KPZ universality class.…

Generative models have proven to be an outstanding tool for representing high-dimensional probability distributions and generating realistic-looking images. An essential characteristic of generative models is their ability to produce…

Absorbing phase transition in restricted exclusion processes are characterized by simple integer exponents. We show that this critical behaviour flows to the directed percolation (DP) universality class when particle conservation is broken…

A range of first-passage percolation type models are believed to demonstrate the related properties of sublinear variance and superdiffusivity. We show that directed last-passage percolation with Gaussian vertex weights has a sublinear…

We examine crossing probabilities and free energies for conformally invariant critical 2-D systems in rectangular geometries, derived via conformal field theory and Stochastic L\"owner Evolution methods. These quantities are shown to…

Percolation theory is usually applied to lattices with a uniform probability p that a site is occupied or that a bond is closed. The more general case, where p is a function of the position x, has received less attention. Previous studies…

Random trajectories of single particles in living cells contain information about the interaction between particles, as well as, with the cellular environment. However, precise consideration of the underlying stochastic properties, beyond…

Random growth models are fundamental objects in modern probability theory, have given rise to new mathematics, and have numerous applications, including tumor growth and fluid flow in porous media. In this article, we introduce some of the…

In this paper we study a two-dimensional directed self-avoiding walk model of a random copolymer in a random emulsion. The copolymer is a random concatenation of monomers of two types, $A$ and $B$, each occurring with density 1/2. The…

In last passage percolation models, the energy of a path is maximized over all directed paths with given endpoints in a random environment, and the maximizing paths are called geodesics. The geodesics and their energy can be scaled so that…

We study the directed polymer model in dimension ${1+1}$ when the environment is heavy-tailed, with a decay exponent $\alpha\in(0,2)$. We give all possible scaling limits of the model in the weak-coupling regime, i.e., when the inverse…

A region of two-dimensional space has been filled randomly with large number of growing circular discs allowing only a `slight' overlapping among them just before their growth stop. More specifically, each disc grows from a nucleation…

Percolation is the simplest fundamental model in statistical mechanics that exhibits phase transitions signaled by the emergence of a giant connected component. Despite its very simple rules, percolation theory has successfully been applied…

For first passage percolation (FPP) on integer lattice with i.i.d. passage time distributions, in order to show existence of semi-infinite geodesics along a fixed direction, one requires unproven assumptions on the limiting shape. We…

The first-passage time (FPT), i.e., the moment when a stochastic process reaches a given threshold value for the first time, is a fundamental mathematical concept with immediate applications. In particular, it quantifies the statistics of…

Grokking -- the delayed transition from memorization to generalization in small algorithmic tasks -- remains poorly understood. We present a geometric analysis of optimization dynamics in transformers trained on modular arithmetic. PCA of…

We consider a generalization of the FKPP equation for the evolution of the spatial density of a single-species population where all the terms are nonlocal. That is, the spatial extension of each process (growth, competition and diffusion)…

We consider a model of first passage percolation (FPP) where the nearest-neighbor edges of the standard two-dimensional Euclidean lattice are equipped with random variables. These variables are i.i.d.\, nonnegative, continuous, and have a…

We consider the Busemann process in planar directed first passage percolation. We extend existing techniques to establish the existence of the process in our setting and determine its distribution in a number of integrable models. As…

We study the contact process on spatially embedded networks, consisting of a regular square lattice with long-range connections. To generate the networks, a long-range connection is randomly added to each node $i$ of a square lattice,…