Shape Theorem Research Articles

Branching random walk with non‐local competition

AbstractWe study the Bolker–Pacala–Dieckmann–Law (BPDL) model of population dynamics in the regime of large population density. The BPDL model is a particle system in which particles reproduce, move randomly in space and compete with each other locally. We rigorously prove global survival as well as a shape theorem describing the asymptotic spread of the population, when the population density is sufficiently large. In contrast to most previous studies, we allow the competition kernel to have an arbitrary, even infinite range, whence the term non‐local competition. This makes the particle system non‐monotone and of infinite‐range dependence, meaning that the usual comparison arguments break down and have to be replaced by a more hands‐on approach. Some ideas in the proof are inspired by works on the non‐local Fisher‐KPP equation, but the stochasticity of the model creates new difficulties.

A shape theorem for exploding sandpiles

A shape theorem for exploding sandpiles

Geodesic length and shifted weights in first-passage percolation

We study first-passage percolation through related optimization problems over paths of restricted length. The path length variable is in duality with a shift of the weights. This puts into a convex duality framework old observations about the convergence of the normalized Euclidean length of geodesics due to Hammersley and Welsh, Smythe and Wierman, and Kesten, and leads to new results about geodesic length and the regularity of the shape function as a function of the weight shift. For points far enough away from the origin, the ratio of the geodesic length and the ℓ 1 \ell ^1 distance to the endpoint is uniformly bounded away from one. The shape function is a strictly concave function of the weight shift. Atoms of the weight distribution generate singularities, that is, points of nondifferentiability, in this function. We generalize to all distributions, directions and dimensions an old singularity result of Steele and Zhang for the planar Bernoulli case. When the weight distribution has two or more atoms, a dense set of shifts produces singularities. The results come from a combination of the convex duality, the shape theorems of the different first-passage optimization problems, and modification arguments.

Covering monotonicity of the limit shapes of first passage percolation on crystal lattices

This paper studies the first passage percolation model on crystal lattices, which is a generalization of that on the cubic lattice. Here, each edge of the graph induced by a crystal lattice is assigned a random passage time, and consideration is given to the behavior of the percolation region $B(t)$, which consists of those vertices that can be reached from the origin within a time $t > 0$. Our first result is the shape theorem, stating that the normalized region $B(t)/t$ converges to some deterministic one, called the limit shape. The second result is the monotonicity of the limit shapes under covering maps. In particular, this provides insight into the limit shape of the cubic first passage percolation model.

A shape theorem and a variational formula for the quenched Lyapunov exponent of random walk in a random potential

Nous prouvons un théorème de forme et déduisons une formule variationnelle pour l’exposant de Lyapunov et la fonction de Green de la marche aléatoire dans un potentiel aléatoire sur un réseau carré de dimension arbitraire et avec un ensemble fini arbitraire des pas possibles. Le potentiel est une fonction d’un environnement stationnaire et du pas de la marche. Ce potentiel est soumis à une hypothèse sur les moments qui est liée à la vitesse de mélange du milieu. Notre cadre comprend les modèles de polymères dirigés et non dirigés, les marche aléatoire dans un environnement aléatoire statique et dynamique, et, dans le cas de température nulle, nos résultats donnent également un théorème de forme et une formule variationnelle pour la constante de temps de la percolation du dernier passage dirigée par site et par arête et de la percolation du premier passage standard.

Intrinsic strong shape of global attractors

We present a proof of the strong shape theorem for global attractors in compact metric spaces using the intrinsic approach to strong shape from [18] which combines continuity over a covering and the corresponding homotopies of second order.

Sharpness of the Phase Transition for the Orthant Model

The orthant model is a directed percolation model on mathbb {Z}^{d}, in which all clusters are infinite. We prove a sharp threshold result for this model: if p is larger than the critical value above which the cluster of 0 is contained in a cone, then the shift from 0 that is required to contain the cluster of 0 in that cone is exponentially small. As a consequence, above this critical threshold, a shape theorem holds for the cluster of 0, as well as ballisticity of the random walk on this cluster.

Stability analysis and Hopf bifurcation based on time delay neurons under electromagnetic fields

Neurons contain a large number of ions inside and outside the cell, and the transmembrane currents formed by the movement of these ions cause membrane potential fluctuations and induce electromagnetism inside and outside the cell. In addition, any change in external electromagnetic fields can cause changes in the membrane potential of the neurons. Therefore, based on the three-dimensional Hindmarsh — Rose (HR) neuron model, a five-dimensional neuron model with time delay is developed in this paper by introducing flux and electric field variables and considering the resulting time delay. First, the Hopf bifurcation theory is used to demonstrate the local stability of the system at the equilibrium point at different time delays. Then, the stability of the Hopf bifurcation and its direction are proved by using the central flow shape theorem. Finally, the existence of the Hopf bifurcation is proved using the phase diagram and the bifurcation diagram, and the effects of several important parameters on the model are investigated by numerical simulations using time series plots, ISI bifurcation plots and two-parameter bifurcation plots. The model is found to be accompanied by chaotic and chaos-free plus-periodic bifurcation structures, mixed-mode discharges and other phenomena. Also, its discharge pattern can be controlled after adding time delay. The results of this paper provide help to the pathogenic mechanism and control of neurological diseases.

A shape theorem for the orthant model

We study a particular model of a random medium, called the orthant model, in general dimensions d≥2. Each site x∈Zd independently has arrows pointing to its positive neighbours x+ei, i=1,…,d with probability p and, otherwise, to its negative neighbours x−ei, i=1,…,d (with probability 1−p). We prove a shape theorem for the set of sites reachable by following arrows, starting from the origin, when p is large. The argument uses subadditivity, as would be expected from the shape theorems arising in the study of first passage percolation. The main difficulty to overcome is that the primary objects of study are not stationary which is a key requirement of the subadditive ergodic theorem.

Quantum mechanics of Plancherel growth

Growth of Young diagrams, equipped with Plancherel measure, follows the automodel equation of Kerov. Using the technology of unitary matrix model we show that such growth process is exactly same as the growth of gap-less phase in Gross-Witten and Wadia (GWW) model. The limit shape of asymptotic Young diagrams corresponds to GWW transition point. Our analysis also offers an alternate proof of limit shape theorem of Vershik-Kerov and Logan-Shepp. Using the connection between unitary matrix model and free Fermi droplet description, we map the Young diagrams in automodel class to different shapes of two dimensional phase space droplets. Quantising these droplets we further set up a correspondence between automodel diagrams and coherent states in the Hilbert space. Thus growth of Young diagrams are mapped to evolution of coherent states in the Hilbert space. Gaussian fluctuations of large N Young diagrams are also mapped to quantum (large N) fluctuations of the coherent states.

The asymptotic shape theorem for the frog model on finitely generated abelian groups

We study the frog model on Cayley graphs of groups with polynomial growth rate D ≥ 3. The frog model is an interacting particle system in discrete time. We consider that the process begins with a particle at each vertex of the graph and only one of these particles is active when the process begins. Each activated particle performs a simple random walk in discrete time activating the inactive particles in the visited vertices. We prove that the activation time of particles grows at least linearly and we show that in the abelian case with any finite generator set the set of activated sites has a limiting shape.

Laplacian growth and sandpiles on the Sierpiński gasket: limit shape universality and exact solutions

We establish quantitative spherical shape theorems for rotor-router aggregation and abelian sandpile growth on the graphical Sierpiński gasket (SG) when particles are launched from the corner vertex. In particular, the abelian sandpile growth problem is exactly solved via a recursive construction of self-similar sandpile tiles. We show that sandpile growth and patterns exhibit a (2\cdot 3^n) -periodicity as a function of the initial mass. Moreover, the cluster explodes – increments by more than 1 in radius – at periodic intervals, a phenomenon not seen on \mathbb{Z}^d or trees. We explicitly characterize all the radial jumps, and use the renewal theorem to prove the scaling limit of the cluster radius, which satisfies a power law modulated by log-periodic oscillations. In the course of our proofs we also establish structural identities of the sandpile groups of subgraphs of SG with two different boundary conditions, notably the corresponding identity elements conjectured by Fairchild, Haim, Setra, Strichartz, and Westura. Our main theorems, in conjunction with recent results of Chen, Huss, Sava-Huss, and Teplyaev, establish SG as a positive example of a state space which exhibits “limit shape universality,” in the sense of Levine and Peres, among the four Laplacian growth models: divisible sandpiles, abelian sandpiles, rotor-router aggregation, and internal diffusion-limited aggregation (IDLA). We conclude the paper with conjectures about radial fluctuations in IDLA on SG, possible extensions of limit shape universality to other state spaces, and related open problems.

Averaging Principle and Shape Theorem for a Growth Model with Memory

AbstractWe present a general approach to study a class of random growth models in n‐dimensional Euclidean space. These models are designed to capture basic growth features that are expected to manifest at the mesoscopic level for several classical self‐interacting processes originally defined at the microscopic scale. It includes once‐reinforced random walk with strong reinforcement, origin‐excited random walk, and a few others, for which the set of visited vertices is expected to form a limiting shape. We prove an averaging principle that leads to such a shape theorem. The limiting shape can be computed in terms of the invariant measure of an associated Markov chain. © 2020 Wiley Periodicals LLC

A Shape Theorem for a One-Dimensional Growing Particle System with a Bounded Number of Occupants per Site

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 derive a limit estimate and an exponential estimate on the fluctuations of the position of the rightmost particle.

Random permutations with logarithmic cycle weights

We consider random permutations on Sn with logarithmic growing cycles weights and study the asymptotic behavior as the length n tends to infinity. We show that the cycle count process converges to a vector of independent Poisson variables and also compute the total variation distance between both processes. Next, we prove a central limit theorem for the total number of cycles. Furthermore we establish a shape theorem and a functional central limit theorem for the Young diagrams associated to random permutations under this measure. We prove these results using tools from complex analysis and combinatorics. In particular we have to apply the method of singularity analysis to generating functions of the form exp((− log(1 − z))k+1) with k ≥ 1, which have not yet been studied in the literature.

Random permutations without macroscopic cycles

We consider uniform random permutations of length $n$ conditioned to have no cycle longer than $n^{\beta }$ with $0<\beta <1$, in the limit of large $n$. Since in unconstrained uniform random permutations most of the indices are in cycles of macroscopic length, this is a singular conditioning in the limit. Nevertheless, we obtain a fairly complete picture about the cycle number distribution at various lengths. Depending on the scale at which cycle numbers are studied, our results include Poisson convergence, a central limit theorem, a shape theorem and two different functional central limit theorems.

Limit Shape of Subpartition-Maximizing Partitions

This is an expository note answering a question posed to us by Richard Stanley, in which we prove a limit shape theorem for partitions of $n$ which maximize the number of subpartitions. The limit shape and the growth rate of the number of subpartitions are explicit. The key ideas are to use large deviations estimates for random walks, together with convex analysis and the Hardy-Ramanujan asymptotics. Our limit shape coincides with Vershik's limit shape for uniform random partitions.

Some recent advances for limit theorems

We present some recent developments for limit theorems in probability theory, illustrating the variety of this field of activity. The recent results we discuss range from Stein’s method, as well as for infinitely divisible distributions as applications of this method in stochastic geometry, to asymptotics for some discrete models. They deal with rates of convergence, functional convergences for correlated random walks and shape theorems for growth models.

A shape theorem for the scaling limit of the IPDSAW at criticality

In this paper we give a complete characterization of the scaling limit of the critical Interacting Partially Directed Self-Avoiding Walk (IPDSAW) introduced in Zwanzig and Lauritzen [J. Chem. Phys. 48 (1968) 3351]. As the system size $L\in\mathbb{N}$ diverges, we prove that the set of occupied sites, rescaled horizontally by $L^{2/3}$ and vertically by $L^{1/3}$ converges in law for the Hausdorff distance toward a nontrivial random set. This limiting set is built with a Brownian motion $B$ conditioned to come back at the origin at $a_{1}$ the time at which its geometric area reaches $1$. The modulus of $B$ up to $a_{1}$ gives the height of the limiting set, while its center of mass process is an independent Brownian motion. Obtaining the shape theorem requires to derive a functional central limit theorem for the excursion of a random walk with Laplace symmetric increments conditioned on sweeping a prescribed geometric area. This result is proven in a companion paper Carmona and Petrelis (2017).

Shape theorem and surface fluctuation for Poisson cylinders

We prove a shape theorem for Poisson cylinders, and give a power-law bound on surface fluctuations. In particular, we show that for any $a \in (1/2, 1)$, conditioned on the origin being in the set of cylinders, if a point belongs to this set and has Euclidean norm below $R$, then this point lies at internal distance less than $R + O(R^{a})$ from the origin.