Laboratoire de Probabilités et Modèles Aléatoires

We prove convergence of solutions of Dirichlet problems and Green's functions on Tutte harmonic embeddings to those of the linearized Monge--Ampère equation $\mathcal{L}_\varphi h=0$. More precisely, we assume that piecewise linear Maxwell--Cremona potentials associated with the embeddings converge to a continuous potential $\varphi$ and the only assumption that we use is the uniform convexity of $\varphi$ or, equivalently, the uniform ellipticity of the operator $\mathcal{L}_\varphi$. Even if $\varphi$ is quadratic, this setup significantly generalizes known results for discrete harmonic functions on orthodiagonal tilings. Motivated by potential applications to the analysis of 2d lattice models on irregular graphs, we also study the situation in which the limits are harmonic in a different complex structure.

Biological control strategies against mosquito-borne diseases--such as the sterile insect technique (SIT), RIDL, and Wolbachia-based releases--require reliable estimates of dispersal and survival of released males. We propose a mechanistic--statistical framework for mark--release--recapture (MRR) data linking an individual-based 2D diffusion model with its reaction--diffusion limit. Inference is based on solving the macroscopic system and embedding it in a Poisson observation model for daily trap counts, with uncertainty quantified via a parametric bootstrap. We validate identifiability using simulated data and apply the model to an urban MRR campaign in El Cano (Havana, Cuba) involving four weekly releases of sterile Aedes aegypti males. The best-supported model suggests a mean life expectancy of about five days and a typical displacement of about 180 m. Unlike empirical fits of survival or dispersal, our mechanistic approach jointly estimates movement, mortality, and capture, yielding biologically interpretable parameters and a principled framework for designing and evaluating SIT-based interventions.

In this article, we propose a novel variable selection approach in the framework of high-dimensional linear models where the columns of the design matrix are highly correlated. It consists in rewriting the initial high-dimensional linear model to remove the correlation between the columns of the design matrix and applying a generalized Elastic Net criterion since it can be seen as an extension of the generalized Lasso. The properties of our approach, called gEN (generalized Elastic Net), are investigated both from a theoretical and a numerical point of view. More precisely, we provide a new condition called GIC (generalized irrepresentable condition), which generalizes the EIC (elastic net irrepresentable condition) of Jia and Yu (2010), under which we prove that our estimator can recover the positions of the null and non null entries of the coefficients when the sample size tends to infinity. We also assess the performance of our methodology using synthetic data and compare it with alternative approaches. Our numerical experiments show that our approach improves the variable selection performance in many cases.

Background: Eosinophilic esophagitis (EoE) is a chronic immune-mediated esophageal disease frequently associated with atopic conditions in children. However, the extent and nature of this association remain unclear. We retrospectively estimated the pr

ABSTRACTIn this note, we analyze the performance of a simple root‐finding algorithm in uniform attachment trees. The leaf‐stripping algorithm recursively removes all leaves of the tree for a carefully chosen number of rounds. We show that, with probability , the set of remaining vertices contains the root and has a size only depending on but not on the size of the tree.

This article studies large and local large deviations for sums of i.i.d. real-valued random variables in the domain of attraction of an α-stable law, α∈(0,2], with emphasis on the case α=2. There are two different scenarios: either the deviation is realised via a collective behaviour with all summands contributing to the deviation (a Gaussian scenario), or a single summand is atypically large and contributes to the deviation (a one-big-jump scenario). Such results are known when α∈(0,2) (large deviations always follow a one big-jump scenario) or when the random variables admit a moment of order 2+δ for some δ>0. We extend these results, including in particular the case where the right tail is regularly varying with index -2 (treating cases with infinite variance in the domain of attraction of the normal law). We identify the threshold for the transition between the Gaussian and the one-big-jump regimes; it is slightly larger when considering local large deviations compared to integral large deviations. Additionally, we complement our results by describing the behaviour of the sum and of the largest summand conditionally on a (local) large deviation, for any α∈(0,2], both in the Gaussian and in the one-big-jump regimes. As an application, we show how our results can be used in the study of condensation phenomenon in the zero-range process at the critical density, extending the range of parameters previously considered in the literature.

We are interested in the geometry of the ``infection tree'' in a stochastic SIR (Susceptible-Infectious-Recovered) model, starting with a single infectious individual. This tree is constructed by drawing an edge between two individuals when one infects the other. We focus on the regime where the infectious period before recovery follows an exponential distribution with rate $1$, and infections occur at a rate $\lambda_{n} \sim \frac{\lambda}{n}$ where $n$ is the initial number of healthy individuals with $\lambda>1$. We show that provided that the infection does not quickly die out, the height of the infection tree is asymptotically $\kappa(\lambda) \log n$ as $n \rightarrow \infty$, where $\kappa(\lambda)$ is a continuous function in $\lambda$ that undergoes a second-order phase transition at $\lambda_{c}\simeq 1.8038$. Our main tools include a connection with the model of uniform attachment trees with freezing and the application of martingale techniques to control profiles of random trees.

Consider some matrix waiting for its coefficients to be written. For each column, sample independently a Bernoulli random variable of some parameter $p$. Seeing all this and possibly using extra randomness, Alice then chooses one spot in each column, in any way she wants. When the Bernoulli random variable of some column is equal to 1, the number 1 is written in the chosen spot. When the Bernoulli random variable of a column is 0, nothing is done on this column. We prove that, using extra randomness, it is possible for Bob to fill the empty entries with well chosen 0's and 1's so that the entries of the matrix are independent Bernoulli random variables of parameter $p$. We investigate various generalisations and variations of this problem, and use this result to revisit and generalise (nonstrict) monotonicity of the percolation threshold $p_c$ with respect to a form of graph-quotienting, namely fibrations. We also use this result to revisit the BK inequality. In a second part, which is independent of the first one, we revisit strict monotonicity of $p_c$ with respect to fibrations, a result that naturally requires more assumptions than its nonstrict counterpart. We reprove the bond-percolation case of the result of Martineau--Severo without resorting to essential enhancements, using couplings instead.

We construct new invariant measures supported on mass level sets for the cubic defocusing nonlinear Schr\"odinger equation in dimensions $1$ and $2$.