Shape Theorem Research Articles

Weak shape theorem in first passage percolation with infinite passage times

Nous considerons le modele standard de percolation de premier passage sur $\mathbb{Z}^{d}$ : nous associons a chaque arete $e$ du graphe un temps de passage $t(e)$ a valeurs dans $[0,+\infty]$, tel que $\mathbb{P}[t(e) p_{c}(d)$. De facon equivalente, nous considerons un modele de percolation de premier passage standard (fini) sur le graphe obtenu par une percolation de Bernoulli sur-critique realisee independamment. Nous prouvons un theoreme de forme faible sans aucune hypothese de moment. Nous prouvons aussi que la constante de temps correspondante est strictement positive si et seulement si $\mathbb{P}[t(e)=0]<p_{c}(d)$.

Convergence Towards an Asymptotic Shape in First-Passage Percolation on Cone-Like Subgraphs of the Integer Lattice

In first-passage percolation on the integer lattice, the Shape Theorem provides precise conditions for convergence of the set of sites reachable within a given time from the origin, once rescaled, to a compact and convex limiting shape. Here, we address convergence towards an asymptotic shape for cone-like subgraphs of the $\Z^d$ lattice, where $d\ge2$. In particular, we identify the asymptotic shapes associated to these graphs as restrictions of the asymptotic shape of the lattice. Apart from providing necessary and sufficient conditions for $L^p$- and almost sure convergence towards this shape, we investigate also stronger notions such as complete convergence and stability with respect to a dynamically evolving environment.

Front progression in the East model

The East model is a one-dimensional, non-attractive interacting particle system with Glauber dynamics, in which a flip is prohibited at a site x if the right neighbour x+1 is occupied. Starting from a configuration entirely occupied on the left half-line, we prove a law of large numbers for the position of the left-most zero (the front), as well as ergodicity of the process seen from the front. For want of attractiveness, the one-dimensional shape theorem is not derived by the usual coupling arguments, but instead by quantifying the local relaxation to the non-equilibrium invariant measure for the process seen from the front. This is the first proof of a shape theorem for a kinetically constrained spin model.

The Varifold Representation of Nonoriented Shapes for Diffeomorphic Registration

In this paper, we address the problem of orientation that naturally arises when representing shapes like curves or surfaces as currents. In the field of computational anatomy, the framework of currents has indeed proved very efficient to model a wide variety of shapes. However, in such approaches, orientation of shapes is a fundamental issue that can lead to several drawbacks in treating certain kind of datasets. More specifically, problems occur with structures like acute pikes because of canceling effects of currents or with data that consists in many disconnected pieces like fiber bundles for which currents require a consistent orientation of all pieces. As a promising alternative to currents, varifolds, introduced in the context of geometric measure theory by F. Almgren, allow the representation of any non-oriented manifold (more generally any non-oriented rectifiable set). In particular, we explain how varifolds can encode numerically non-oriented objects both from the discrete and continuous point of view. We show various ways to build a Hilbert space structure on the set of varifolds based on the theory of reproducing kernels. We show that, unlike the currents' setting, these metrics are consistent with shape volume (theorem 4.1) and we derive a formula for the variation of metric with respect to the shape (theorem 4.2). Finally, we propose a generalization to non-oriented shapes of registration algorithms in the context of Large Deformations Metric Mapping (LDDMM), which we detail with a few examples in the last part of the paper.

A note on geodesics for supercritical continuum percolation

In this paper, we obtain moderate deviations for the graph distance in supercritical continuum percolation and the corresponding shape theorem. Using the moderate deviations, we get the fluctuation exponents for the geodesics, which are not yet obtained for lattice first passage percolation. These results have applications in communication networks.

Block characters of the symmetric groups

A block character of a finite symmetric group is a positive definite function which depends only on the number of cycles in a permutation. We describe the cone of block characters by identifying its extreme rays, and find relations of the characters to descent representations and the coinvariant algebra of ${\mathfrak{S}}_{n}$ . The decomposition of extreme block characters into the sum of characters of irreducible representations gives rise to certain limit shape theorems for random Young diagrams. We also study counterparts of the block characters for the infinite symmetric group ${\mathfrak{S}}_{\infty}$ , along with their connection to the Thoma characters of the infinite linear group GL ?(q) over a Galois field.

Asymptotic shape for the contact process in random environment

The aim of this article is to prove asymptotic shape theorems for the contact process in stationary random environment. These theorems generalize known results for the classical contact process. In particular, if $H_{t}$ denotes the set of already occupied sites at time $t$, we show that for almost every environment, when the contact process survives, the set $H_{t}/t$ almost surely converges to a compact set that only depends on the law of the environment. To this aim, we prove a new almost subadditive ergodic theorem.

On the internal distance in the interlacement set

We prove a shape theorem for the internal (graph) distance on the interlacement set $\mathcal{I}^u$ of the random interlacement model on $\mathbb Z^d$, $d\ge 3$. We provide large deviation estimates for the internal distance of distant points in this set, and use these estimates to study the internal distance on the range of a simple random walk on a discrete torus.

Rotor-Router Aggregation on the Comb

We prove a shape theorem for rotor-router aggregation on the comb, for a specific initial rotor configuration and clockwise rotor sequence for all vertices. Furthermore, as an application of rotor-router walks, we describe the harmonic measure of the rotor-router aggregate and related shapes, which is useful in the study of other growth models on the comb. We also identify the shape for which the harmonic measure is uniform. This gives the first known example where the rotor-router cluster has non-uniform harmonic measure, and grows with different speeds in different directions.

A shape theorem for Riemannian first-passage percolation

Riemannian first-passage percolation is a continuum model, with a distance function arising from a random Riemannian metric in Rd. Our main result is a shape theorem for this model, which says that large balls under this metric converge to a deterministic shape under rescaling. As a consequence, we show that smooth random Riemannian metrics are geodesically complete with probability of 1.

The asymptotic shape theorem for generalized first passage percolation

We generalize the asymptotic shape theorem in first passage percolation on $\mathbb{Z}^d$ to cover the case of general semimetrics. We prove a structure theorem for equivariant semimetrics on topological groups and an extended version of the maximal inequality for $\mathbb{Z}^d$-cocycles of Boivin and Derriennic in the vector-valued case. This inequality will imply a very general form of Kingman's subadditive ergodic theorem. For certain classes of generalized first passage percolation, we prove further structure theorems and provide rates of convergence for the asymptotic shape theorem. We also establish a general form of the multiplicative ergodic theorem of Karlsson and Ledrappier for cocycles with values in separable Banach spaces with the Radon--Nikodym property.

A shape theorem for the spread of an infection

In [KSb] we studied the following model for the spread of a rumor or infection: There is a “gas” of so-called A-particles, each of which performs a continuous time simple random walk on Z d , with jump rate DA. We assume that “just before the start” the number of A-particles at x, NA(x, 0−), has a mean μA Poisson distribution and that the NA(x, 0−) ,x ∈ Z d , are independent. In addition, there are B-particles which perform continuous time simple random walks with jump rate DB. We start with a finite number of B-particles in the system at time 0. The positions of these initial B-particles are arbitrary, but they are nonrandom. The B-particles move independently of each other. The only interaction occurs when a B-particle and an A-particle coincide; the latter instantaneously turns into a B-particle. [KSb] gave some basic estimates for the growth of the set � B(t ): ={x ∈ Z d :a B-particle visits x during [0 ,t ]}.

Dimer packings with gaps and electrostatics

Fisher and Stephenson conjectured in 1963 that the correlation function (defined by dimer packings) of two unit holes on the square lattice is rotationally invariant in the limit of large separation between the holes. We consider the same problem on the hexagonal lattice, extend it to an arbitrary finite collection of holes, and present an explicit conjectural answer. In recent work we managed to prove this conjecture in two fairly general cases. The quantity giving the answer can be regarded as the exponential of the negative of the two-dimensional electrostatic energy of a system of charges naturally associated with the holes. We further develop this analogy to electrostatics by presenting two different natural ways to define a field in our setup, and showing that both lead to the electric field, in the limit of large separations between the holes. For one of the fields, this is also stated as a limit shape theorem for random surfaces, with the continuum limit being a sum of helicoids. We conclude by explaining the relationship of our results to previous results in the physics literature on spin correlations in the Ising model.

An oriented competition model on $Z_+^2$

We consider a two-type oriented competition model on the first quadrant of the two-dimensional integer lattice. Each vertex of the space may contain only one particle of either Red type or Blue type. A vertex flips to the color of a randomly chosen southwest nearest neighbor at exponential rate 2. At time zero there is one Red particle located at $$. The main result is a partial shape theorem: Denote by $R (t)$ and $B (t)$ the red and blue regions at time $t$. Then (i) eventually the upper half of the unit square contains no points of $B (t)/t$, and the lower half no points of $R (t)/t$; and (ii) with positive probability there are angular sectors rooted at $$ that are eventually either red or blue. The second result is contingent on the uniform curvature of the boundary of the corresponding Richardson shape.

Multitype shape theorems for first passage percolation models

A Euclidean first passage percolation model describing the competing growth between k different types of infection is considered. We focus on the long-time behavior of this multitype growth process and we derive multitype shape results related to its morphology.

Multitype shape theorems for first passage percolation models

A Euclidean first passage percolation model describing the competing growth between k different types of infection is considered. We focus on the long-time behavior of this multitype growth process and we derive multitype shape results related to its morphology.

Limit shapes for random square Young tableaux

Our main result is a limit shape theorem for the two-dimensional surface defined by a uniform random n × n square Young tableau. The analysis leads to a calculus of variations minimization problem that resembles the minimization problems studied by Logan–Shepp, Vershik–Kerov, and Cohn–Larsen–Propp. We solve this problem by developing a general technique for solving variational problems of this kind. An extension to rectangular Young tableaux is also given. We also apply the main result to show that the location of a particular entry in the tableau is in the limit governed by a semicircle distribution, and to the study of extremal Erdös–Szekeres permutations, namely permutations of the numbers 1 , 2 , … , n 2 whose longest monotone subsequence is of length n.

Large deviations and phase transition for random walks in random nonnegative potentials

We establish large deviation principles and phase transition results for both quenched and annealed settings of nearest-neighbor random walks with constant drift in random nonnegative potentials on Z d . We complement the analysis of M.P.W. Zerner [Directional decay of the Green’s function for a random nonnegative potential on Z d , Ann. Appl. Probab. 8 (1996) 246–280], where a shape theorem on the Lyapunov functions and a large deviation principle in absence of the drift are achieved for the quenched setting.

Internal Diffusion Limited Aggregation on Discrete Groups Having Exponential Growth

The internal diffusion limited aggregation (DLA) has been introduced by Diaconis and Fulton [Rend. Sem. Mat. Univ. Pol. Torino 49, 95–119 (1991)]. It is a growth model defined on an infinite set and associated to a Markov chain on this set. We focus here on sets which are finitely generated groups with exponential growth. We prove a shape theorem for the internal DLA on such groups associated to symmetric random walks. For that purpose, we introduce a new distance associated to the Green function, which happens to have some interesting properties. In the case of homogeneous trees, we also get the right order for the fluctuations of that model around its limiting shape.

The spread of a rumor or infection in a moving population

We consider the following interacting particle system: There is a “gas” of particles, each of which performs a continuous-time simple random walk on ℤd, with jump rate DA. These particles are called A-particles and move independently of each other. They are regarded as individuals who are ignorant of a rumor or are healthy. We assume that we start the system with NA(x,0−) A-particles at x, and that the NA(x,0−),x∈ℤd, are i.i.d., mean-μA Poisson random variables. In addition, there are B-particles which perform continuous-time simple random walks with jump rate DB. We start with a finite number of B-particles in the system at time 0. B-particles are interpreted as individuals who have heard a certain rumor or who are infected. The B-particles move independently of each other. The only interaction is that when a B-particle and an A-particle coincide, the latter instantaneously turns into a B-particle. We investigate how fast the rumor, or infection, spreads. Specifically, if B̃(t):={x∈ℤd: a B-particle visits x during [0,t]} and B(t)=B̃(t)+[−1/2,1/2]d, then we investigate the asymptotic behavior of B(t). Our principal result states that if DA=DB (so that the A- and B-particles perform the same random walk), then there exist constants 0<Ci<∞ such that almost surely $\mathcal{C}(C_{2}t)\subset B(t)\subset \mathcal{C}(C_{1}t)$ for all large t, where $\mathcal{C}(r)=[-r,r]^{d}$. In a further paper we shall use the results presented here to prove a full “shape theorem,” saying that t−1B(t) converges almost surely to a nonrandom set B0, with the origin as an interior point, so that the true growth rate for B(t) is linear in t. If DA≠DB, then we can only prove the upper bound $B(t)\subset \mathcal{C}(C_{1}t)$ eventually.