Multiple Phase Transitions in Long‐Range First‐Passage Percolation on Square Lattices

Abstract

We consider a model of long-range first-passage percolation on the ddimensional square lattice Z d in which any two distinct vertices x; y 2 Z d are connected by an edge having exponentially distributed passage time with mean kx yk˛C o.1/ , where˛> 0 is a fixed parameter and k k is the`1-norm on Z d . We analyze the asymptotic growth rate of the set B t , which consists of all x 2 Z d such that the first-passage time between the origin 0 and x is at most t as t ! 1. We show that depending on the values of˛there are four growth regimes: (i) instantaneous growth for˛< d , (ii) stretched exponential growth for˛2 .d; 2d /, (iii) superlinear growth for˛2 .2d; 2d C 1/, and finally (iv) linear growth for˛> 2d C 1 like the nearest-neighbor first-passage percolation model corresponding to˛D 1. LONG-RANGE FIRST-PASSAGE PERCOLATION207 used certain versions of LRFPP for modeling biological invasion of species [17,24,42,51]. Along with many other factors they use dispersal kernels r. / with heavy tails as part of their models for dispersal mechanisms of biological objects (such as seeds, pollen, fungi, etc.). However, most of their conclusions are based on simulations in two-dimensional grid and nonrigorous heuristics. In [17], followed by [42], the authors recognized two phases of spatiotemporal behavior, which they call long-distance dispersal and short-distance dispersal, based on whether the second moment of the dispersal kernel is infinite or finite. They argue that under finite second-moment conditions (short-distance dispersal regime) the growth behavior of the region reachable within time t is same as that in nearest-neighbor (or finiterange) FPP. On the other hand, the authors in [24] recognized one additional phase, which they call medium-distance dispersal, but they didn't specify where the transitions between different phases occur. As we will prove here, the situation is much more delicate, and there are at least four distinct phases (with three critical points in between) depending on the heavy tail index of the dispersal kernel.Aldous [2] considered communication of continuously arriving information through a finite agent network in a certain game-theoretic setup. In one of the cases, where the network topology is a two-dimensional discrete torus and the communication cost between any two agents is a nondecreasing function of the euclidean distance between them, the main technical tool to understand the time evolution of the fraction of informed agents is the analysis of the LRFPP model, which we propose here, on a large two-dimensional discrete torus. Aldous proposed a simplified version of this LRFPP model, which he named short-long FPP, in which agent network topology is a discrete torus, each pair of nearest-neighbor agents communicate at rate 1, and all other pairs of agents communicate at a rate that depends only on the size of the torus regardless of the distance between the agents. The continuous analogue of the short-long FPP model has been analyzed rigorously on a (two-dimensional) real torus [18] and...

Polynomial chaos and scaling limits of disordered systems

“…However, it is far from obvious whether A α,β (·) converges to some asymptotic process A α,∞ (·) asβ → ∞ and whether such a process describes some universality class for long-range random polymers, last passage percolation and growth models, generalizing the the so-called Airy process obtained for α = 2. Besides a very recent work on the limit shapes of long-range first-passage percolation model [CD13], long-range polymer type models do not appear to have been studied systematically before.…”

Acceleration of evolutionary spread by long-range dispersal

“…Potentially, even very rare jumps over exceptionally large distances could be important (14). If this is the case, then the stochastic nature of the jumps that drive the dynamics will be essential.Although evolutionary spread with long-range jumps has been simulated stochastically in a number of biological contexts (6,8,(19)(20)(21)(22)(23), few analytic results have been obtained on the ensuing stochastic dynamics (19,(24)(25)(26). Most analyses have resorted to deterministic approximations (27-35), which are successful for describing both the local and global dispersal limits.…”

“…Although evolutionary spread with long-range jumps has been simulated stochastically in a number of biological contexts (6,8,(19)(20)(21)(22)(23), few analytic results have been obtained on the ensuing stochastic dynamics (19,(24)(25)(26). Most analyses have resorted to deterministic approximations (27)(28)(29)(30)(31)(32)(33)(34)(35), which are successful for describing both the local and global dispersal limits.…”

From sectors to speckles: The impact of long-range migration on gene surfing

“…The kernel exponent µ quantifies the heaviness of the tail of the dispersal kernel, with higher values corresponding to jump distributions that fall off more steeply with distance. Power-law kernels of this type encompass a broad swath of population growth dynamics, ranging from effectively well-mixed (µ → 0) to effectively diffusive (µ > d + 1) [38,39]. We will henceforth refer to the population size as the number of demes, and generations are defined by the average time between migration events out of an occupied deme.…”