Related papers: Internal DLA and the Gaussian free field

In a recent paper (Bogoyavlenskiy V A 2002 \JPA \textbf{35} 2533), an algorithm aiming to generate isotropic clusters of the on-lattice diffusion-limited aggregation (DLA) model was proposed. The procedure consists of aggregation…

We study the scaling limits of three different aggregation models on Z^d: internal DLA, in which particles perform random walks until reaching an unoccupied site; the rotor-router model, in which particles perform deterministic analogues of…

We employ the recently introduced conformal iterative construction of Diffusion Limited Aggregates (DLA) to study the multifractal properties of the harmonic measure. The support of the harmonic measure is obtained from a dynamical process…

In this paper we prove an analogue in the discrete setting of \Bbb Z^d, of the spherical maximal theorem for \Bbb R^d. The methods used are two-fold: the application of certain "sampling" techniques, and ideas arising in the study of the…

This paper studies the limits of a spatial random field generated by uniformly scattered random sets, as the density $\lambda$ of the sets grows to infinity and the mean volume $\rho$ of the sets tends to zero. Assuming that the volume…

Let F be an orthonormal basis of weight 2 cusp forms on Gamma_0(N). We show that various weighted averages of special values L(f \tensor chi, 1) over f in F are equal to 4 pi + O(N^{-1 + epsilon}). A previous result of Duke gives an error…

We analyze nonequilibrium fluctuations of the averaging process on $\mathbb T_\varepsilon^d$, a continuous degenerate Gibbs sampler running over the edges of the discrete $d$-dimensional torus. We show that, if we start from a smooth…

We study the pairwise annihilation process $A+A\to$ inert of a number of random walkers, which originally are localized in a small region in space. The size of the colony and the typical distance between particles increases with time and,…

Laplacian growth, associated to the diffusion-limited aggregation (DLA) model or the more general dielectric-breakdown model (DBM), is a fundamental out-of-equilibrium process that generates structures with characteristic…

We study the limiting fluctuations of the number of sign and level-set clusters of the Gaussian free field on $\mathbb{Z}^d$, $d \ge 3$, that are contained in a large domain. In dimension $d \ge 4$ we prove that the fluctuations are…

We consider the problem of efficiently learning mixtures of a large number of spherical Gaussians, when the components of the mixture are well separated. In the most basic form of this problem, we are given samples from a uniform mixture of…

We performed extensive numerical simulation of diffusion-limited aggregation in two dimensional channel geometry. Contrary to earlier claims, the measured fractal dimension D = 1.712 +- 0.002 and its leading correction to scaling are the…

We study the high excursion probability of a centered Gaussian field on a square. Writing \(\sigma\) and \(r\) for its standard deviation and correlation function, we assume that \(\sigma\) has a unique maximum at the corner…

We analyze the sample complexity of full-batch Gradient Descent (GD) in the setup of non-smooth Stochastic Convex Optimization. We show that the generalization error of GD, with common choice of hyper-parameters, can be $\tilde \Theta(d/m +…

A recent line of works, initiated by Russo and Xu, has shown that the generalization error of a learning algorithm can be upper bounded by information measures. In most of the relevant works, the convergence rate of the expected…

We propose a novel framework for exploring weak and $L_2$ generalization errors of algorithms through the lens of differential calculus on the space of probability measures. Specifically, we consider the KL-regularized empirical risk…

We study sample-to-sample fluctuations in a critical two-dimensional Ising model with quenched random ferromagnetic couplings. Using replica calculations in the renormalization group framework we derive explicit expressions for the…

We study the generalized diffusion-limited aggregates (DLA), with two seeds placed at distance d lattice units and investigate the probability p(d) that the patterns generated from those seeds get connected. In this model, one can vary the…

We investigate in this paper the error on the cluster redshift estimate as a function of: (1) the number of galaxy structures along the line of sight; (2) the morphology of the clusters (regular/substructures); (3) the nature of the…

The debate on the correlation properties of galaxy structures has having an increasing interest during the last year. In this lecture we discuss the claims of different authors who have criticized our approach and results. In order to have…