Related papers: Growth of uniform infinite causal triangulations
We discuss uniform infinite causal triangulations and equivalence to the size biased branching process measure - the critical Galton-Watson branching process distribution conditioned on non-extinction. Using known results from the theory of…
A construction as a growth process for sampling of the uniform infinite planar triangulation (UIPT), defined in a previous paper, is given. The construction is algorithmic in nature, and is an efficient method of sampling a portion of the…
We establish a connection between the uniform infinite planar triangulation and some critical time-reversed branching process. This allows to find a scaling limit for the principal boundary component of a ball of radius R for large R (i.e.…
Consider a branching random walk in which the offspring distribution and the moving law both depend on an independent and identically distributed random environment indexed by the time.For the normalised counting measure of the number of…
We consider infinite particle system on the positive half-line moving independently of each other. When a particle hits the boundary it immediately disappears, and the boundary moves to the right on some fixed quantity (particle size). We…
The purpose of the present work is twofold. First, we develop the theory of general self-similar growth-fragmentation processes by focusing on martingales which appear naturally in this setting and by recasting classical results for…
A recent preprint (arXiv:1402.2632) introduced a growth procedure for planar maps, whose almost sure limit is "the uniform infinite 3-connected planar map". A classical construction of Brooks, Smith, Stone and Tutte (1940) associates a…
We consider a growing planar network where a tip grows at constant speed, branches at constant rate and inactivates when it meets a branch already created. We only consider here orthogonal branching occurring always in the same direction.…
We are interested in the cycles obtained by slicing at all heights random Boltzmann triangulations with a simple boundary. We establish a functional invariance principle for the lengths of these cycles, appropriately rescaled, as the size…
We use a growth procedure for binary trees due to Luczak and Winkler, a bijection between binary trees and irreducible quadrangulations of the hexagon due to Fusy, Poulalhon and Schaeffer, and the classical angular mapping between…
We introduce and study the uniform infinite planar quadrangulation (UIPQ) with a boundary via an extension of the construction of arXiv:1201.1052. We then relate this object to its simple boundary analog using a pruning procedure. This…
In an evolutionary system in which the rules of mutation are local in nature, the number of possible outcomes after $m$ mutations is an exponential function of $m$ but with a rate that depends only on the set of rules and not the size of…
Many growth processes lead to intriguing stochastic patterns and complex fractal structures which exhibit local scale invariance properties. Such structures can often be described effectively by space-time trajectories of interacting…
We consider a Markov evolution of lozenge tilings of a quarter-plane and study its asymptotics at large times. One of the boundary rays serves as a reflecting wall. We observe frozen and liquid regions, prove convergence of the local…
This article is concerned with pointwise growth and spreading speeds in systems of parabolic partial differential equations. Several criteria exist for quantifying pointwise growth rates. These include the location in the complex plane of…
Looptrees have recently arisen in the study of critical percolation on the uniform infinite planar triangulation. Here we consider random infinite looptrees defined as the local limit of the looptree associated with a critical…
We present a coalescent process where three particles merge at each coagulation step. Using a random walk representation, we prove duality with a fragmentation process, whose fragmentation law we specify explicitly. Furthermore, we give a…
Random-cluster measures on infinite regular trees are studied in conjunction with a general type of `boundary condition', namely an equivalence relation on the set of infinite paths of the tree. The uniqueness and non-uniqueness of…
Time evolutions whose infinitesimal generator is a fractional time derivative arise generally in the long time limit. Such fractional time evolutions are considered here for random walks. An exact relationship is given between the…
It is well-known that the excursions of a one-dimensional diffusion process can be studied by considering a certain Riccati equation associated with the process. We show that, in many cases of interest, the Riccati equation can be solved in…