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Amine Asselah

Publications and source records attributed to Amine Asselah.

At least 19 recordsLinked to original sources
arXiv2025

Derivative formula for capacities

We obtain a derivative formula for various notions of capacity. Namely we identify the second order term in the asymptotic expansion of the capacity of a union of two sets, as their distance goes to infinity. Our result applies to the usual Newtonian capacity in the setting of random walks on the Euclidean lattice, to the family of Bessel-Riesz capacities, and to the Branching capacity, which has been introduced recently by Zhu [9] in connection with critical Branching random walks. On the other hand, the result remains open for the notion of capacity in the setting of percolation, which is introduced in a companion paper, but serves as a motivation, as it would have some interesting consequences there.

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arXiv2025

Internal Diffusion Limited Aggregation with Critical Branching Random Walks

Internal Diffusion Limited Aggregation is an interacting particle system that describes the growth of a random cluster governed by the boundary harmonic measure seen from an internal point. Our paper studies IDLA in $\mathbb{Z}^d$ driven by critical branching random walks. We prove that, unlike classical IDLA, this process exhibits a phase transition in the dimension. More precisely, we establish the existence of a spherical shape theorem in dimension $d\geq 3$ and the absence of a spherical shape theorem for $d \leq 2$. Our bounds on the inner and outer worst deviations are of polynomial nature, which we expect to be a feature of this model.

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arXiv2025

Capacity in high dimensional percolation

We introduce a notion of capacity for high dimensional critical percolation by showing that for any finite set $A$, the suitably rescaled probability that the cluster of $z$ intersects $A$ converges as $\|z\|\to\infty$. This can be viewed as a generalisation of the asymptotic of the two point function and we call the limit the p-capacity of $A$. We next show that the probability that the Incipient Infinite Cluster of $z$ intersects the set $A$ appropriately normalised is also of order the p-capacity of $A$ as $\|z\|\to\infty$. We conjecture that the p-capacity is of the same order as the $(d-4)$-Bessel-Riesz capacity and in support of this we estimate the p-capacity of balls. As a byproduct of our techniques we give a simpler proof of the one-arm exponent of Kozma and Nachmias for dimensions 8 and higher and as long as the two point function asymptotic holds. Our proofs make use of a new large deviations bound on the pioneers, that is the number of points on the boundary of a box which are part of the cluster of the origin restricted to this box.

math.PR↗
arXiv2024

On the intersection of critical percolation clusters and other tree-like random graphs

We study intersection properties of two or more independent tree-like random graphs. Our setting encompasses critical, possibly long range, Bernoulli percolation clusters, incipient infinite clusters, as well as critical branching random walk ranges. We obtain sharp excess deviation bounds on the number of intersection points of two or more clusters, under minimal assumption on the two-point function. The proof are based on new bounds on the n-point function, in case of critical percolation, and on the joint moments of local times of branching random walks

math.PR↗
arXiv2024

Time spent in a ball by a critical branching random walk

We study a critical branching random walk on $\mathbb Z^d$. We focus on the tail of the time spent in a ball, and our study, in dimension four and higher, sheds new light on the recent result of Angel, Hutchcroft and Jarai, in particular on the special features of the critical dimension four. Finally, we analyse the number of walks transported by the branching random walk on the boundary of a distant ball.

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arXiv2024

The Critical Density for Activated Random Walks is always less than 1

Activated Random Walks, on $\mathbb{Z}^d$ for any $d\geqslant 1$, is an interacting particle system, where particles can be in either of two states: active or frozen. Each active particle performs a continuous-time simple random walk during an exponential time of parameter $λ$, after which it stays still in the frozen state, until another active particle shares its location, and turns it instantaneously back into activity. This model is known to have a phase transition, and we show that the critical density, controlling the phase transition, is less than one in any dimension and for any value of the sleep rate $λ$. We provide upper bounds for the critical density in both the small $λ$ and large $λ$ regimes.

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arXiv2023

Branching random walks and Minkowski sum of random walks

We show that the range of a critical branching random walk conditioned to survive forever and the Minkowski sum of two independent simple random walk ranges are intersection-equivalent in any dimension $d\ge 5$, in the sense that they hit any finite set with comparable probability, as their common starting point is sufficiently far away from the set to be hit. Furthermore, we extend a discrete version of Kesten, Spitzer and Whitman's result on the law of large numbers for the volume of a Wiener sausage. Here, the sausage is made of the Minkowski sum of $N$ independent simple random walk ranges in $\mathbb{Z}^d$, with $d>2N$, and of a finite set $A\subset \mathbb{Z}^d$. When properly normalised the volume of the sausage converges to a quantity equivalent to the capacity of $A$ with respect to the kernel $K(x,y)=(1+\|x-y\|)^{2N-d}$. As a consequence, we establish a new relation between capacity and {\it branching capacity}.

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arXiv2023

Local times and capacity for transient branching random walks

We consider branching random walks on the Euclidean lattice in dimensions five and higher. In this non-Markovian setting, we first obtain a relationship between the equilibrium measure and Green's function, in the form of an approximate last passage decomposition. Secondly, we obtain exponential moment bounds for functionals of the branching random walk, under optimal condition. As a corollary we obtain an approximate variational characterisation of the branching capacity. We finally derive upper bounds involving the branching capacity for the tail of the time spent in an arbitrary finite collection of balls. This generalises the results of [AHJ] and [AS22] for $d\geq 5$. For random walks, the analogous tail estimates have been instrumental tools for tackling deviations problems on the range, related to folding of the walk.

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arXiv2020

Extracting subsets maximizing capacity and Folding of Random Walks

We prove that in any finite set of $\mathbb Z^d$ with $d\ge 3$, there is a subset whose capacity and volume are both of the same order as the capacity of the initial set. As an application we obtain estimates on the probability of {\it covering uniformly} a finite set, and characterize some {\it folding} events, under optimal hypotheses. For instance, knowing that a region of space has an {\it atypically high occupation density} by some random walk, we show that this random region is most likely ball-like

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arXiv2020

Deviations for the Capacity of the Range of a Random Walk

We obtain estimates for large and moderate deviations for the capacity of the range of a random walk on $\mathbb{Z}^d$, in dimension $d\ge 5$, both in the upward and downward directions. The results are analogous to those we obtained for the volume of the range in two companion papers [AS17, AS19]. Interestingly, the main steps of the strategy we developed for the latter apply in this seemingly different setting, yet the details of the analysis are different

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arXiv2020

Moderate deviations for the range of a transient random walk. II

We obtain sharp upper and lower bounds for the moderate deviations of the volume of the range of a random walk in dimension five and larger. Our results encompass two regimes: a Gaussian regime for small deviations, and a stretched exponential regime for larger deviations. In the latter regime, we show that conditioned on the moderate deviations event, the walk folds a small part of its range in a ball-like subset. Also, we provide new path properties, in dimension three as well. Besides the key role Newtonian capacity plays in this study, we introduce two original ideas, of general interest, which strengthen the approach developed in \cite{AS}.

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arXiv2020

Large Deviations for Intersections of Random Walks

We prove a Large Deviations Principle for the number of intersections of two independent infinite-time ranges in dimension five and more, improving upon the moment bounds of Khanin, Mazel, Shlosman and Sina{ï} [KMSS94]. This settles, in the discrete setting, a conjecture of van den Berg, Bolthausen and den Hollander [BBH04], who analyzed this question for the Wiener sausage in finite-time horizon. The proof builds on their result (which was resumed in the discrete setting by Phetpradap [Phet12]), and combines it with a series of tools that were developed in recent works of the authors [AS17, AS19a, AS20]. Moreover, we show that most of the intersection occurs in a single box where both walks realize an occupation density of order one.

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arXiv2019

On outer fluctuations for internal DLA

We had established inner and outer fluctuation for the internal DLA cluster when all walks are launched from the origin. In obtaining the outer fluctuation, we had used a deep lemma of Jerison, Levine and Sheffield, which estimate roughly the possibility of fingering, and had provided a simple proof using an interesting estimate for crossing probability for a simple random walk. The application of the crossing probability to the fingering for the internal DLA cluster contains a flaw discovered recently, that we correct in this note. We take the opportunity to make a self-contained exposition.

math.PR↗
arXiv2018

Moderate deviations for the range of a transient random walk: path concentration

We study downward deviations of the boundary of the range of a transient walk on the Euclidean lattice. We describe the optimal strategy adopted by the walk in order to shrink the boundary of its range. The technics we develop apply equally well to the range, and provide pathwise statements for the {\it Swiss cheese} picture of Bolthausen, van den Berg and den Hollander \cite{BBH}.

math.PR↗
arXiv2018

On the nature of the Swiss cheese in dimension 3

We study scenarii linked with the Swiss cheese picture in dimension three obtained when two random walks are forced to meet often, or when one random walk is forced to squeeze its range. In the case of two random walks, we show that they most likely meet in a region of optimal density. In the case of one random walk, we show that a small range is reached by a strategy uniform in time. Both results rely on an original inequality estimating the cost of visiting sparse sites, and in the case of one random walk on the precise Large Deviation Principle of van den Berg, Bolthausen and den Hollander, including their sharp estimates of the rate functions in the neighborhood of the origin

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arXiv2017

Ergodicity of a system of interacting random walks with asymmetric interaction

We study N interacting random walks on the positive integers. Each particle has drift δ towards infinity, a reflection at the origin, and a drift towards particles with lower positions. This inhomogeneous mean field system is shown to be ergodic only when the interaction is strong enough. We focus on this latter regime, and point out the effect of piles of particles, a phenomenon absent in models of interacting diffusion in continuous space.

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