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Gregorio R. Moreno Flores

Publications and source records attributed to Gregorio R. Moreno Flores.

14 recordsLinked to original sources
arXiv2023

Directed Polymers in Complex-Valued Random Environments on the Tree

We consider a model of directed polymers on regular trees with complex-valued random weights introduced by Cook and Derrida [CD90] and studied mathematically by Derrida, Evans and Speer [DES93]. In addition to the usual weak-disorder and strong-disorder regimes, the phase diagram of the model contains a third region due to the effects of the random phases that cannot be observed in the model with positive weights. In this work, we extend the results of [DES93] in two directions. First, we remove the hypothesis on the independence of the random radii and random phases of the environment from most of the phase diagram, and second, under mild assumptions on the law of the environment, we strengthen the convergence of the free energy (shown to hold in probability in [DES93]) to an almost sure convergence. Along the way, many of the arguments of [DES93] are simplified.

math.PR↗
arXiv2023

One-dimensional Discrete Dirac Operators in a Decaying Random Potential II: Clock, Schrödinger and Sine statistics

We consider one-dimensional discrete Dirac models in vanishing random environments. In a previous work [6], we showed that these models exhibit a rich phase diagram in terms of their spectrum as a function of the rate of decay of the random potential. This article is devoted to their spectral statistics. We show that the rescaled spectrum converges to the clock process for fast decay and to the Schrödinger/Sine processes from random matrix theory for critical decay. This way, we recover all the regimes previously identified for the Anderson model in a similar context [25]. Poisson statistics, which should appear in the model with slow decay, are left as an open problem. The core of the proof consists in a suitable scaling limit for the Prüfer phase and monotonicity arguments, yielding an alternative to the approach of [25]. For one of the models, we also obtain the scaling limit of the Prüfer radii and discuss the consequences for the limiting shape of the eigenfunctions.

math-ph↗
arXiv2021

Scaling limit of stationary coupled Sasamoto-Spohn models

We introduce a family of stationary coupled Sasamoto-Spohn models and show that, in the weakly asymmetric regime, they converge to the energy solution of coupled Burgers equations. Moreover, we show that any system of coupled Burgers equations satisfying the so-called trilinear condition ensuring stationarity can be obtained as the scaling limit of a suitable system of coupled Sasamoto-Spohn models. The core of our proof, which avoids the use of spectral gap estimates, consists in a second order Boltzmann-Gibbs principle for the discrete model.

math.PR↗
arXiv2020

Aging for the stationary Kardar-Parisi-Zhang equation and related models

We study the aging property for stationary models in the KPZ universality class. In particular, we show aging for the stationary KPZ fixed point, the Cole-Hopf solution to the stationary KPZ equation, the height function of the stationary TASEP, last-passage percolation with boundary conditions and stationary directed polymers in the intermediate disorder regime. All of these models are shown to display a universal aging behavior characterized by the rate of decay of their correlations. As a comparison, we show aging for models in the Edwards-Wilkinson universality class where a different decay exponent is obtained. A key ingredient to our proofs is a characteristic of space-time stationarity - covariance-to-variance reduction - which allows to deduce the asymptotic behavior of the correlations of two space-time points by the one of the variances at one point. We formulate several open problems.

math.PR↗
arXiv2020

One-dimensional Discrete Anderson Model in a Decaying Random Potential: from a.c. Spectrum to Dynamical Localization

We consider a one-dimensional Anderson model where the potential decays in average like $n^{-α}$, $α>0$. This simple model is known to display a rich phase diagram with different kinds of spectrum arising as the decay rate $α$ varies. We review an article of Kiselev, Last and Simon where the authors show a.c. spectrum in the super-critical case $α>\frac12$, a transition from singular continuous to pure point spectrum in the critical case $α=\frac12$, and dense pure point spectrum in the sub-critical case $α<\frac12$. We present complete proofs of the cases $α\ge\frac12$ and simplify some arguments along the way. We complement the above result by discussing the dynamical aspects of the model. We give a simple argument showing that, despite of the spectral transition, transport occurs for all energies for $α=\frac12$. Finally, we discuss a theorem of Simon on dynamical localization in the sub-critical region $α<\frac12$. This implies, in particular, that the spectrum is pure point in this regime.

math-ph↗
arXiv2020

One-dimensional Discrete Dirac Operators in a Decaying Random Potential I: Spectrum and Dynamics

We study the spectrum and dynamics of a one-dimensional discrete Dirac operator in a random potential obtained by damping an i.i.d. environment with an envelope of type $n^{-α}$ for $α>0$. We recover all the spectral regimes previously obtained for the analogue Anderson model in a random decaying potential, namely: absolutely continuous spectrum in the super-critical region $α>\frac12$; a transition from pure point to singular continuous spectrum in the critical region $α=\frac12$; and pure point spectrum in the sub-critical region $α<\frac12$. From the dynamical point of view, delocalization in the super-critical region follows from the RAGE theorem. In the critical region, we exhibit a simple argument based on lower bounds on eigenfunctions showing that no dynamical localization can occur even in the presence of point spectrum. Finally, we show dynamical localization in the sub-critical region by means of the fractional moments method and provide control on the eigenfunctions.

math-ph↗
arXiv2019

Stationary Directed Polymers and Energy Solutions of the Burgers Equation

We consider the stationary O'Connell-Yor model of semi-discrete directed polymers in a Brownian environment in the intermediate disorder regime and show convergence of the increments of the log-partition function to the energy solutions of the stochastic Burgers equation. The proof does not rely on the Cole-Hopf transform and avoids the use of spectral gap estimates for the discrete model. The key technical argument is a second-order Boltzmann-Gibbs principle.

math.PR↗
arXiv2018

Scaling of the Sasamoto-Spohn model in equilibrium

We prove the convergence of the Sasamoto-Spohn model in equilibrium to the energy solution of the stochastic Burgers equation on the whole line. The proof, which relies on the second order Boltzmann-Gibbs principle, follows the approach of \cite{GJS} and does not use any spectral gap argument.

math.PR↗
arXiv2018

Random polymers on the complete graph

Consider directed polymers in a random environment on the complete graph of size $N$. This model can be formulated as a product of i.i.d. $N\times N$ random matrices and its large time asymptotics is captured by Lyapunov exponents and the Furstenberg measure. We detail this correspondence, derive the long-time limit of the model and obtain a co-variant distribution for the polymer path. Next, we observe that the model becomes exactly solvable when the disorder variables are located on edges of the complete graph and follow a totally asymmetric stable law of index $α\in (0,1)$. Then, a certain notion of mean height of the polymer behaves like a random walk and we show that the height function is distributed around this mean according to an explicit law. Large $N$ asymptotics can be taken in this setting, for instance, for the free energy of the system and for the invariant law of the polymer height with a shift. Moreover, we give some perturbative results for environments which are close to the totally asymmetric stable laws.

math.PR↗
arXiv2014

On the (strict) positivity of solutions of the stochastic heat equation

We give a new proof of the fact that the solutions of the stochastic heat equation, started with non-negative initial conditions, are strictly positive at positive times. The proof uses concentration of measure arguments for discrete directed polymers in Gaussian environments, originated in M. Talagrand's work on spin glasses and brought to directed polymers by Ph. Carmona and Y. Hu. We also get slightly improved bounds on the lower tail of the solutions of the stochastic heat equation started with a delta initial condition.

math.PR↗
arXiv2013

Fluctuation exponents for directed polymers in the intermediate disorder regime

We compute the fluctuation exponents for a solvable model of one-dimensional directed polymers in random environment in the intermediate regime. This regime corresponds to taking the inverse temperature to zero with the size of the system. The exponents satisfy the KPZ scaling relation and coincide with physical predictions. In the critical case, we recover the fluctuation exponents of the Cole-Hopf solution of the KPZ equation in equilibrium and close to equilibrium.

math.PR↗
arXiv2012

Edge scaling of the β-Jacobi ensemble

We study the scaling limit of the spectrum of the β-Jacobi ensemble at the soft-edge and hard-edge for general values of β. We show that the limiting point processes correspond respectively to the stochastic Airy and Bessel point processes introduced in Ramirez-Rider-Virag and Ramirez-Rider.

math.PR↗
arXiv2010

Asymmetric directed polymers in random environments

The model of Brownian Percolation has been introduced as an approximation of discrete last-passage percolation models close to the axis. It allowed to compute some explicit limits and prove fluctuation theorems for these, based on the relations between the Brownian percolation and random matrices. Here, we present two approaches that allow to treat discrete asymmetric models of directed polymers. In both cases, the behaviour is universal, meaning that the results do not depend on the precise law of the environment as long as it satisfies some natural moment assumptions. First, we establish an approximation of asymmetric discrete directed polymers in random environments at very high temperature by a continuous-time directed polymers model in a Brownian environment, much in the same way than the last passage percolation case. The key ingredient is a strong embedding argument developed by Kómlos, Major and Túsnady. Then, we study the partition function of a $1+1$-dimensional directed polymer in a random environment with a drift tending to infinity. We give an explicit expression for the free energy based on known asymptotics for last-passage percolation and compute the order of the fluctuations of the partition function. We conjecture that the law of the properly rescaled fluctuations converges to the GUE Tracy-Widom distribution.

math.PR↗