Related papers: Random growth models: shape and convergence rate
The talk presented at ICMP 97 focused on the scaling limits of critical percolation models, and some other systems whose salient features can be described by collections of random lines. In the scaling limit we keep track of features seen…
In this survey article, a variety of systems modeling tumor growth are discussed. In accordance with the hallmarks of cancer, the described models incorporate the primary characteristics of cancer evolution. Specifically, we focus on…
Percolation is a model for random damage to a network. It is one of the simplest models that displays a phase transition: when the network is severely damaged, it falls apart in many small connected components, while if the damage is light,…
This paper is a survey of various results and techniques in first passage percolation, a random process modeling a spreading fluid on an infinite graph. The latter half of the paper focuses on the connection between first passage…
In this work we are considering the behavior of the limit shape of Young diagrams associated to random permutations on the set $\{1,\dots,n\}$ under a particular class of multiplicative measures. Our method is based on generating functions…
This study delves into the exploration of the limiting shape theorem for subadditive processes on finitely generated groups with polynomial growth, commonly referred to as virtually nilpotent groups. Investigating the algebraic structures…
We consider a class of growth models and models of turbulence based on the randomly stirred fluid. The similarity between the predictions of these models, noted a decade earlier, is understood on the basis of a stochastic quantization…
A random growth lattice filling model of percolation with touch and stop growth rule is developed and studied numerically on a two dimensional square lattice. Nucleation centers are continuously added one at a time to the empty sites and…
We consider a one-dimensional discrete-space birth process with a bounded number of particle per site. Under the assumptions of the finite range of interaction, translation invariance, and non-degeneracy, we prove a shape theorem. We also…
Random graph models are used to describe the complex structure of real-world networks in diverse fields of knowledge. Studying their behavior and fitting properties are still critical challenges, that in general, require model specific…
We study a random growth model on $\R^d$ introduced by Deijfen. This is a continuous first-passage percolation model. The growth occurs by means of spherical outbursts with random radii in the infected region. We aim at finding conditions…
We study the contact process in a dynamical random environment defined on the vertices and edges of a graph. For a broad class of processes, we establish an asymptotic shape theorem for the set H_t, which represents the vertices that have…
The random diffusion model is a continuum model for a conserved scalar density field driven by diffusive dynamics where the bare diffusion coefficient is density dependent. We generalize the model from one with a sharp wavenumber cutoff to…
Convergence rate estimates in limit theorems for sums of independent random variables are considered.
We examine a lattice model of tumor growth where survival of tumor cells depends on the supplied nutrients. When such a supply is random, the extinction of tumors belongs to the directed percolation universality class. However, when the…
We introduce a model for a growing random graph based on simultaneous reproduction of the vertices. The model can be thought of as a generalisation of the reproducing graphs of Southwell and Cannings and Bonato et al to allow for a random…
Initiation and development of a malignant tumor is a complex phenomenon that has critical stages determining its long time behavior. This phenomenon is mathematically described by means of various models: from simple heuristic models to…
We review the recent fast progress in statistical physics of evolving networks. Interest has focused mainly on the structural properties of random complex networks in communications, biology, social sciences and economics. A number of giant…
The goal of these lectures is to review some mathematical aspects of random tree models used in evolutionary biology to model gene trees or species trees. We start with stochastic models of tree shapes (finite trees without edge lengths),…
This study delves into first-passage percolation on random geometric graphs in the supercritical regime, where the graphs exhibit a unique infinite connected component. We investigate properties such as geodesic paths, moderate deviations,…