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Alejandro Ramírez

Publications and source records attributed to Alejandro Ramírez.

7 recordsLinked to original sources
arXiv2022

Cyclic products and optimal traps in cyclic birth and death chains

A birth-death chain is a discrete-time Markov chain on the integers whose transition probabilities $p_{i,j}$ are non-zero if and only if $|i-j|=1$. We consider birth-death chains whose birth probabilities $p_{i,i+1}$ form a periodic sequence, so that $p_{i,i+1}=p_{i \mod m}$ for some $m$ and $p_0,\ldots,p_{m-1}$. The trajectory $(X_n)_{n=0,1,\ldots}$ of such a chain satisfies a strong law of large numbers and a central limit theorem. We study the effect of reordering the probabilities $p_0,\ldots,p_{m-1}$ on the velocity $v=\lim_{n\to\infty} X_n/n$. The sign of $v$ is not affected by reordering, but its magnitude in general is. We show that for Lebesgue almost every choice of $(p_0,\ldots,p_{m-1})$, exactly $(m-1)!/2$ distinct speeds can be obtained by reordering. We make an explicit conjecture of the ordering that minimises the speed, and prove it for all $m\leq 7$. This conjecture is implied by a purely combinatorial conjecture that we think is of independent interest.

math.PR↗
arXiv2021

Balanced Excited Random Walk in Two Dimensions

We give non-trivial upper and lower bounds on the range of the so-called Balanced Excited Random Walk in two dimensions, and verify a conjecture of Benjamini, Kozma and Schapira. To the best of our knowledge these are the first non-trivial results for this 2-dimensional model

math.PR↗
arXiv2021

The effect of disorder on quenched and averaged large deviations for random walks in random environments: boundary behavior

For a random walk in a uniformly elliptic and i.i.d. environment on $\mathbb Z^d$ with $d \geq 4$, we show that the quenched and annealed large deviations rate functions agree on any compact set contained in the boundary $\partial \mathbb{D}:=\{ x \in \mathbb R^d : |x|_1 =1\}$ of their domain which does not intersect any of the $(d-2)$-dimensional facets of $\partial \mathbb{D}$, provided that the disorder of the environment is~low~enough. As a consequence, we obtain a simple explicit formula for both rate functions on $\partial \mathbb{D}$ at low disorder. In contrast to previous works, our results do not assume any ballistic behavior of the random walk and are not restricted to neighborhoods of any given point (on the boundary $\partial \mathbb{D}$). In addition, our~results complement those in [BMRS19], where, using different methods, we investigate the equality of the rate functions in the interior of their domain. Finally, for a general parametrized family of environments, we~show that the strength of disorder determines a phase transition in the equality of both rate functions, in the sense that for each $x \in \partial \mathbb{D}$ there exists $\varepsilon_x$ such that the two rate functions agree at $x$ when the disorder is smaller than $\varepsilon_x$ and disagree when its larger. This further reconfirms the idea, introduced in [BMRS19], that the disorder of the environment is in general intimately related with the equality of the rate functions.

math.PR↗
arXiv2019

End-to-end Cloud Segmentation in High-Resolution Multispectral Satellite Imagery Using Deep Learning

Segmenting clouds in high-resolution satellite images is an arduous and challenging task due to the many types of geographies and clouds a satellite can capture. Therefore, it needs to be automated and optimized, specially for those who regularly process great amounts of satellite images, such as governmental institutions. In that sense, the contribution of this work is twofold: We present the CloudPeru2 dataset, consisting of 22,400 images of 512x512 pixels and their respective hand-drawn cloud masks, as well as the proposal of an end-to-end segmentation method for clouds using a Convolutional Neural Network (CNN) based on the Deeplab v3+ architecture. The results over the test set achieved an accuracy of 96.62%, precision of 96.46%, specificity of 98.53%, and sensitivity of 96.72% which is superior to the compared methods.

cs.CV↗
arXiv2016

Fluctuations of the front in a one-dimensional model for the spread of an infection

We study the following microscopic model of infection or epidemic reaction: red and blue particles perform independent nearest-neighbor continuous-time symmetric random walks on the integer lattice $\mathbb{Z}$ with jump rates $D_R$ for red particles and $D_B$ for blue particles, the interaction rule being that blue particles turn red upon contact with a red particle. The initial condition consists of i.i.d. Poisson particle numbers at each site, with particles at the left of the origin being red, while particles at the right of the origin are blue. We are interested in the dynamics of the front, defined as the rightmost position of a red particle. For the case $D_R=D_B$, Kesten and Sidoravicius established that the front moves ballistically, and more precisely that it satisfies a law of large numbers. Their proof is based on a multi-scale renormalization technique, combined with approximate sub-additivity arguments. In this paper, we build a renewal structure for the front propagation process, and as a corollary we obtain a central limit theorem for the front when $D_R=D_B$. Moreover, this result can be extended to the case where $D_R>D_B$, up to modifying the dynamics so that blue particles turn red upon contact with a site that has previously been occupied by a red particle. Our approach extends the renewal structure approach developed by Comets, Quastel and Ram\'{ı}rez for the so-called frog model, which corresponds to the $D_B=0$ case.

math.PR↗
arXiv2008

Large Deviations of the Front in a one dimensional model of $X+Y \to 2X$

We investigate the probabilities of large deviations for the position of the front in a stochastic model of the reaction $X+Y \to 2X$ on the integer lattice in which $Y$ particles do not move while $X$ particles move as independent simple continuous time random walks of total jump rate $2$. For a wide class of initial conditions, we prove that a large deviations principle holds and we show that the zero set of the rate function is the interval $[0,v]$, where $v$ is the velocity of the front given by the law of large numbers. We also give more precise estimates for the rate of decay of the slowdown probabilities. Our results indicate a gapless property of the generator of the process as seen from the front, as it happens in the context of nonlinear diffusion equations describing the propagation of a pulled front into an unstable state.

math.PR↗