arXiv subjects

Anne Fey

Publications and source records attributed to Anne Fey.

11 recordsLinked to original sources

arXiv2012

Critical densities in sandpile models with quenched or annealed disorder

We discuss various critical densities in sandpile models. The stationary density is the average expected height in the stationary state of a finite-volume model; the transition density is the critical point in the infinite-volume counterpart. These two critical densities were generally assumed to be equal, but this has turned out to be wrong for deterministic sandpile models. We show they are not equal in a quenched version of the Manna sandpile model either. In the literature, when the transition density is simulated, it is implicitly or explicitly assumed to be equal to either the so-called threshold density or the so-called critical activity density. We properly define these auxiliary densities, and prove that in certain cases, the threshold density is equal to the transition density. We extend the definition of the critical activity density to infinite volume, and prove that in the standard infinite volume sandpile, it is equal to 1. Our results should bring some order in the precise relations between the various densities.

math-ph↗

arXiv2011

Metastability threshold for anisotropic bootstrap percolation in three dimensions

In this paper we analyze several anisotropic bootstrap percolation models in three dimensions. We present the order of magnitude for the metastability threshold for a fairly general class of models. In our proofs we use an adaptation of the technique of dimensional reduction. We find that the order of the metastability threshold is generally determined by the "easiest growth direction" in the model. In contrast to the anisotropic bootstrap percolation in two dimensions, in three dimensions the order of the metatstability threshold for anisotropic bootstrap percolation can be equal to that of isotropic bootstrap percolation.

math-ph↗

arXiv2010

Limiting shapes for a non-abelian sandpile growth model and related cellular automata

We present limiting shape results for a non-abelian variant of the abelian sandpile growth model (ASGM), some of which have no parallel in the ASGM. One of our limiting shapes is an octagon. In our model, mass spreads from the origin by the toppling rule in Zhang's sandpile model. Previously, several limiting shape results have been obtained for the ASGM using abelianness and monotonicity as main tools. As both properties fail for our model, we use a new proof technique: in our main proof, we introduce several cellular automata to mimic our growth model.

math.CO↗

arXiv2010

Driving sandpiles to criticality and beyond

A popular theory of self-organized criticality relates driven dissipative systems to systems with conservation. This theory predicts that the stationary density of the abelian sandpile model equals the threshold density of the fixed-energy sandpile. We refute this prediction for a wide variety of underlying graphs, including the square grid. Driven dissipative sandpiles continue to evolve even after reaching criticality. This result casts doubt on the validity of using fixed-energy sandpiles to explore the critical behavior of the abelian sandpile model at stationarity.

cond-mat.stat-mech↗

arXiv2010

The approach to criticality in sandpiles

A popular theory of self-organized criticality relates the critical behavior of driven dissipative systems to that of systems with conservation. In particular, this theory predicts that the stationary density of the abelian sandpile model should be equal to the threshold density of the corresponding fixed-energy sandpile. This "density conjecture" has been proved for the underlying graph Z. We show (by simulation or by proof) that the density conjecture is false when the underlying graph is any of Z^2, the complete graph K_n, the Cayley tree, the ladder graph, the bracelet graph, or the flower graph. Driven dissipative sandpiles continue to evolve even after a constant fraction of the sand has been lost at the sink. These results cast doubt on the validity of using fixed-energy sandpiles to explore the critical behavior of the abelian sandpile model at stationarity.

math.PR↗

arXiv2009

Growth Rates and Explosions in Sandpiles

We study the abelian sandpile growth model, where n particles are added at the origin on a stable background configuration in Z^d. Any site with at least 2d particles then topples by sending one particle to each neighbor. We find that with constant background height h <= 2d-2, the diameter of the set of sites that topple has order n^{1/d}. This was previously known only for h<d. Our proof uses a strong form of the least action principle for sandpiles, and a novel method of background modification. We can extend this diameter bound to certain backgrounds in which an arbitrarily high fraction of sites have height 2d-1. On the other hand, we show that if the background height 2d-2 is augmented by 1 at an arbitrarily small fraction of sites chosen independently at random, then adding finitely many particles creates an explosion (a sandpile that never stabilizes).

math.CO↗

arXiv2009

Stabilizability and percolation in the infinite volume sandpile model

We study the sandpile model in infinite volume on $\mathbb{Z}^d$. In particular, we are interested in the question whether or not initial configurations, chosen according to a stationary measure $μ$, are $μ$-almost surely stabilizable. We prove that stabilizability does not depend on the particular procedure of stabilization we adopt. In $d=1$ and $μ$ a product measure with density $ρ=1$ (the known critical value for stabilizability in $d=1$) with a positive density of empty sites, we prove that $μ$ is not stabilizable. Furthermore, we study, for values of $ρ$ such that $μ$ is stabilizable, percolation of toppled sites. We find that for $ρ>0$ small enough, there is a subcritical regime where the distribution of a cluster of toppled sites has an exponential tail, as is the case in the subcritical regime for ordinary percolation.

math.PR↗

arXiv2008

Large deviations for eigenvalues of sample covariance matrices, with applications to mobile communication systems

We study sample covariance matrices of the form $W=\frac 1n C C^T$, where $C$ is a $k\times n$ matrix with i.i.d. mean zero entries. This is a generalization of so-called Wishart matrices, where the entries of $C$ are independent and identically distributed standard normal random variables. Such matrices arise in statistics as sample covariance matrices, and the high-dimensional case, when $k$ is large, arises in the analysis of DNA experiments. We investigate the large deviation properties of the largest and smallest eigenvalues of $W$ when either $k$ is fixed and $n\to \infty$, or $k_n\to \infty$ with $k_n=o(n/\log\log{n})$, in the case where the squares of the i.i.d. entries have finite exponential moments. Previous results, proving a.s. limits of the eigenvalues, only require finite fourth moments. Our most explicit results for $k$ large are for the case where the entries of $C$ are $\pm1$ with equal probability. We relate the large deviation rate functions of the smallest and largest eigenvalue to the rate functions for independent and identically distributed standard normal entries of $C$. This case is of particular interest, since it is related to the problem of the decoding of a signal in a code division multiple access system arising in mobile communication systems. In this example, $k$ plays the role of the number of users in the system, and $n$ is the length of the coding sequence of each of the users. Each user transmits at the same time and uses the same frequency, and the codes are used to distinguish the signals of the separate users. The results imply large deviation bounds for the probability of a bit error due to the interference of the various users.

math.PR↗

arXiv2008

Uniqueness of the stationary distribution and stabilizability in Zhang's sandpile model

We show that Zhang's sandpile model (N,[a,b]) on N sites and with uniform additions on [a,b] has a unique stationary measure for all 0 <= a < b <= 1. This generalizes earlier results where this was shown in some special cases. We define the infinite volume Zhang's sandpile model in dimension d >= 1, in which topplings occur according to a Markov toppling process, and we study the stabilizability of initial configurations chosen according to some measure μ. We show that for a stationary ergodic measure μwith density ρ, for all ρ< 1/2, μis stabilizable; for all ρ>= 1, μis not stabilizable; for 1/2 <= ρ< 1, when ρis near to 1/2 or 1, both possibilities can occur.

math-ph↗

arXiv2007

Limiting shapes for deterministic centrally seeded growth models

We study the rotor router model and two deterministic sandpile models. For the rotor router model in $\mathbb{Z}^d$, Levine and Peres proved that the limiting shape of the growth cluster is a sphere. For the other two models, only bounds in dimension 2 are known. A unified approach for these models with a new parameter $h$ (the initial number of particles at each site), allows to prove a number of new limiting shape results in any dimension $d \geq 1$. For the rotor router model, the limiting shape is a sphere for all values of $h$. For one of the sandpile models, and $h=2d-2$ (the maximal value), the limiting shape is a cube. For both sandpile models, the limiting shape is a sphere in the limit $h \to -\infty$. Finally, we prove that the rotor router shape contains a diamond.

math.PR↗

arXiv2007

A probabilistic approach to Zhang's sandpile model

The current literature on sandpile models mainly deals with the abelian sandpile model (ASM) and its variants. We treat a less known - but equally interesting - model, namely Zhang's sandpile. This model differs in two aspects from the ASM. First, additions are not discrete, but random amounts with a uniform distribution on an interval $[a,b]$. Second, if a site topples - which happens if the amount at that site is larger than a threshold value $E_c$ (which is a model parameter), then it divides its entire content in equal amounts among its neighbors. Zhang conjectured that in the infinite volume limit, this model tends to behave like the ASM in the sense that the stationary measure for the system in large volumes tends to be peaked narrowly around a finite set. This belief is supported by simulations, but so far not by analytical investigations. We study the stationary distribution of this model in one dimension, for several values of $a$ and $b$. When there is only one site, exact computations are possible. Our main result concerns the limit as the number of sites tends to infinity, in the one-dimensional case. We find that the stationary distribution, in the case $a \geq E_c/2$, indeed tends to that of the ASM (up to a scaling factor), in agreement with Zhang's conjecture. For the case $a=0$, $b=1$ we provide strong evidence that the stationary expectation tends to $\sqrt{1/2}$.

math-ph↗