arXiv subjects

Ricardo Misturini

Publications and source records attributed to Ricardo Misturini.

7 recordsLinked to original sources

arXiv2023

How many real zeros does a random Dirichlet series have?

Let $F(σ)=\sum_{n=1}^\infty \frac{X_n}{n^σ}$ be a random Dirichlet series where $(X_n)_{n\in\mathbb{N}}$ are independent standard Gaussian random variables. We compute in a quantitative form the expected number of zeros of $F(σ)$ in the interval $[T,\infty)$, say $\mathbb{E} N(T,\infty)$, as $T\to1/2^+$. We also estimate higher moments and with this we derive exponential tails for the probability that the number of zeros in the interval $[T,1]$, say $N(T,1)$, is large. We also consider almost sure lower and upper bounds for $N(T,\infty)$. And finally, we also prove results for another class of random Dirichlet series, e.g., when the summation is restricted to prime numbers.

math.NT↗

arXiv2023

From ABC to KPZ

We study the equilibrium fluctuations of an interacting particle system evolving on the discrete ring with $N\in\mathbb N$ points, denoted by $\mathbb T_N$, and with three species of particles that we name $A,B$ and $C$, but such that at each site there is only one particle. We prove that proper choices of density fluctuation fields (that match those from nonlinear fluctuating hydrodynamics theory) associated to the (two) conserved quantities converge, in the limit $N\to\infty$, to a system of stochastic partial differential equations, that can either be the Ornstein-Uhlenbeck equation or the Stochastic Burgers equation. To understand the cross interaction between the two conserved quantities, we derive a general version of the Riemann-Lebesgue lemma which is of independent interest.

math.PR↗

arXiv2023

Full $\Gamma$-expansion of reversible Markov chains level two large deviations rate functionals

Let $\Xi_n \subset \mathbb R^d$, $n\ge 1$, be a sequence of finite sets and consider a $\Xi_n$-valued, irreducible, reversible, continuous-time Markov chain $(X^{(n)}_t:t\ge 0)$. Denote by $\mathscr P(\mathbb R^d) $ the set of probability measures on $\mathbb R^d$ and by $I_n\colon \mathscr P(\mathbb R^d) \to [0,+\infty)$ the level two large deviations rate functional for $X^{(n)}_t$ as $t\to\infty$. We present a general method, based on tools used to prove the metastable behaviour of Markov chains, to derive a full expansion of $I_n$ expressing it as $I_n = I^{(0)} \,+\, \sum_{1\le p\le q} (1/\theta^{(p)}_n)\, I^{(p)}$, where $I^{(p)}\colon \mathscr P(\mathbb R^d) \to [0,+\infty]$ represent rate functionals independent of $n$ and $\theta^{(p)}_n$ sequences such that $\theta^{(1)}_n \to\infty$, $\theta^{(p)}_n / \theta^{(p+1)}_n \to 0$ for $1\le p< q$. The speed $\theta^{(p)}_n$ corresponds to the time-scale at which the Markov chains $X^{(n)}_t$ exhibits a metastable behavior, and the $I^{(p-1)}$ zero-level sets to the metastable states. To illustrate the theory we apply the method to random walks in potential fields.

math.PR↗

arXiv2022

Hydrodynamics for the ABC model with slow/fast boundary

In this article, we consider the ABC model in contact with slow/fast reservoirs. In this model, there is at most one particle per site, which can be of type $α\in\{A,B,C\}$ and particles exchange positions in the discrete set of points $\{1,\cdots, N-1\}$ with a weakly asymmetric rate that depends on the type of particles involved in the exchange mechanism. At the boundary points $x=1, N-1$ particles can be injected or removed with a rate that depends on the type of particles involved. We prove that the hydrodynamic limit, in the diffusive time scale, is given by a system of non-linear coupled equation with several boundary conditions, that depend on the strength of the reservoir's action.

math.PR↗

arXiv2021

The boundary driven zero-range process

We study the asymptotic behaviour of the symmetric zero-range process in the finite lattice $\{1,\ldots, N-1\}$ with slow boundary, in which particles are created at site $1$ or annihilated at site $N\!-\!1$ with a rate proportional to $N^{-θ}$, for $θ\geq 1$. We present the invariant measure for this model and obtain the hydrostatic limit. In order to understand the asymptotic behaviour of the spatial-temporal evolution of this model under the diffusive scaling, we start to analyze the hydrodynamic limit, exploiting attractiveness as an essential ingredient. We obtain that the hydrodynamic equation has boundary conditions that depend on the value of $θ$.

math.PR↗

arXiv2020

Law of the iterated logarithm for a random Dirichlet series

Let $(X_n)_{n\in \mathbb{N}}$ be a sequence of i.i.d. random variables with distribution $\mathbb P(X_1=1)=\mathbb P(X_1=-1)=1/2$. Let $F(σ)=\sum_{n=1}^\infty X_nn^{-σ}$. We prove that the following holds almost surely \begin{equation*} \limsup_{σ\to 1/2^+}\frac{F(σ)}{\sqrt{2\mathbb E F(σ)^2\log\log \mathbb E F(σ)^2}}=1. \end{equation*}

math.PR↗

arXiv2014

Evolution of the ABC model among the segregated configurations in the zero-temperature limit

We consider the ABC model on a ring in a strongly asymmetric regime. The main result asserts that the particles almost always form three pure domains (one of each species) and that this segregated shape evolves, in a proper time scale, as a Brownian motion on the circle, which may have a drift. This is, to our knowledge, the first proof of a zero-temperature limit for a non-reversible dynamics whose invariant measure is not explicitly known.

math.PR↗