arXiv subjects

Giampaolo Cristadoro

Publications and source records attributed to Giampaolo Cristadoro.

At least 19 recordsLinked to original sources

arXiv2026

Double-transform Tauberian method for precise large deviations

In many stochastic models, the observables of interest are naturally encoded in double transforms (e.g., Laplace transforms) that couple spatial and temporal variables. Notably, the double transform often provides the only analytically tractable starting point for the study of processes with correlated increments or path constraints. We extend the Tauberian approach for precise large deviations of stochastic processes belonging to the domain of attraction of spectrally positive stable laws, previously developed for single-variable Laplace--Stieltjes transforms [9], to the bivariate setting. This methodology provides a direct route to asymptotic behaviour that is otherwise difficult to characterize using single-transform techniques. As illustrative examples, we derive precise large deviations for random sums with increments correlated to the stopping time and for path-dependent observables of random walks constrained to remain positive.

math.PR↗

arXiv2026

Precise large deviations through a uniform Tauberian theorem

We derive a large deviation principle for families of random variables in the basin of attraction of spectrally positive stable distributions by proving a uniform version of the Tauberian theorem for Laplace-Stieltjes transforms. The main advantage of this method is that it can be easily applied to cases that are beyond the reach of the techniques currently used in the literature. Notable examples include large deviations for random walks with long-ranged memory kernels, as well as for randomly stopped sums where the random time $\mathrm N$ is either not concentrated around its expectation or has an infinite mean. The method reveals the role of the characteristic function when Cramér's condition is violated and provides a unified approach within regular variation.

math.PR↗

arXiv2025

First-passage statistics of random walks: a general approach via Riemann-Hilbert problems

We study first-passage statistics for one-dimensional random walks $S_n$ with independent and identically distributed jumps starting from the origin. We focus on the joint distribution of the first-passage time $τ_b$ and first-passage position $S_{τ_b}$ beyond a threshold $b\geq0$, as well as the distribution of $S_n$ for the walks that do not cross $b$ up to step $n$. By solving suitable Riemann-Hilbert problems, we are able to obtain exact and semi-explicit general formulae for the quantities of interest. Notably, such formulae are written solely in terms of the characteristic function of the jumps. In contrast with previous results, our approach is universally valid, applicable to both continuous and discrete, symmetric and asymmetric jump distributions. We complement our theoretical findings with explicit examples.

cond-mat.stat-mech↗

arXiv2025

Ladder costs for random walks in Lévy random media

We consider a random walk $Y$ moving on a Lévy random medium, namely a one-dimensional renewal point process with inter-distances between points that are in the domain of attraction of a stable law. The focus is on the characterization of the law of the first-ladder height $Y_{\mathcal{T}}$ and length $L_{\mathcal{T}}(Y)$, where $\mathcal{T}$ is the first-passage time of $Y$ in $\mathbb{R}^+$. The study relies on the construction of a broader class of processes, denoted Random Walks in Random Scenery on Bonds (RWRSB) that we briefly describe. The scenery is constructed by associating two random variables with each bond of $\mathbb{Z}$, corresponding to the two possible crossing directions of that bond. A random walk $S$ on $\mathbb{Z}$ with i.i.d increments collects the scenery values of the bond it traverses: we denote this composite process the RWRSB. Under suitable assumptions, we characterize the tail distribution of the sum of scenery values collected up to the first exit time $\mathcal{T}$. This setting will be applied to obtain results for the laws of the first-ladder length and height of $Y$. The main tools of investigation are a generalized Spitzer-Baxter identity, that we derive along the proof, and a suitable representation of the RWRSB in terms of local times of the random walk $S$. All these results are easily generalized to the entire sequence of ladder variables.

math.PR↗

arXiv2024

Optimal conditions for first passage of jump processes with resetting

We investigate the first passage time beyond a barrier located at $b\geq0$ of a random walk with independent and identically distributed jumps, starting from $x_0=0$. The walk is subject to stochastic resetting, meaning that after each step the evolution is restarted with fixed probability $r$. We consider a resetting protocol that is an intermediate situation between a random walk ($r=0$) and an uncorrelated sequence of jumps all starting from the origin ($r=1$), and derive a general condition for determining when restarting the process with $0<r<1$ is more efficient than restarting after each jump. If the mean first passage time of the process in absence of resetting is larger than that of the sequence of jumps, this condition is sufficient to establish the existence of an optimal $0<r^*<1$ that represents the best strategy, outperforming both $r=0$ and $r=1$. Our findings are discussed by considering two important examples of jump processes, for which we draw the phase diagram illustrating the regions of the parameter space where resetting with some $0<r^*<1$ is optimal.

cond-mat.stat-mech↗

arXiv2024

Optimizing Leapover Lengths of Lévy Flights with Resetting

We consider a one-dimensional search process under stochastic resetting conditions. A target is located at $b\geq0$ and a searcher, starting from the origin, performs a discrete-time random walk with independent jumps drawn from a heavy-tailed distribution. Before each jump, there is a given probability $r$ of restarting the walk from the initial position. The efficiency of a "myopic search" - in which the search stops upon crossing the target for the first time - is usually characterized in terms of the first-passage time $τ$. On the other hand, great relevance is encapsulated by the leapover length $l = x_τ - b$, which measures how far from the target the search ends. For symmetric heavy-tailed jump distributions, in the absence of resetting the average leapover is always infinite. Here we show instead that resetting induces a finite average leapover $\ell_b(r)$ if the mean jump length is finite. We compute exactly $\ell_b(r)$ and determine the condition under which resetting allows for nontrivial optimization, i.e., for the existence of $r^*$ such that $\ell_b(r^*)$ is minimal and smaller than the average leapover of the single jump.

cond-mat.stat-mech↗

arXiv2023

Internal-wave billiards in trapezoids and similar tables

We call internal-wave billiard the dynamical system of a point particle that moves freely inside a planar domain (the table) and is reflected by its boundary according to this rule: reflections are standard Fresnel reflections but with the pretense that the boundary at any collision point is either horizontal or vertical (relative to a predetermined direction representing gravity). These systems are point particle approximations for the motion of internal gravity waves in closed containers, hence the name. For a class of tables similar to rectangular trapezoids, but with the slanted leg replaced by a general curve with downward concavity, we prove that the dynamics has only three asymptotic regimes: (1) there exist a global attractor and a global repellor, which are periodic and might coincide; (2) there exists a beam of periodic trajectories, whose boundary (if any) comprises an attractor and a repellor for all the other trajectories; (3) all trajectories are dense (that is, the system is minimal). Furthermore, in the prominent case where the table is an actual trapezoid, we study the sets in parameter space relative to the three regimes. We prove in particular that the set for (1) has positive measure (giving a rigorous proof of the existence of Arnol'd tongues for internal-wave billiards), whereas the sets for (2) and (3) are non-empty but have measure zero.

math.DS↗

arXiv2022

Recurrence times, waiting times and universal entropy production estimators

The universal typical-signal estimators of entropy and cross entropy based on the asymptotics of recurrence and waiting times play an important role in information theory. Building on their construction, we introduce and study universal typical-signal estimators of entropy production in the context of nonequilibrium statistical mechanics of one-sided shifts over finite alphabets.

math-ph↗

arXiv2022

Maximal escape rate for shifts

We consider the shift transformation on the space of infinite sequences over a finite alphabet endowed with the invariant product measure, and examine the presence of a \emph{hole} on the space. The holes we study are specified by the sequences that do not contain a given finite word as initial sub-string. The measure of the set of sequences that do not fall into the hole in the first $n$ iterates of the shift is known to decay exponentially with $n$, and its exponential rate is called \emph{escape rate}. In this paper we provide a complete characterization of the holes with maximal escape rate. In particular we show that, contrary to the case of equiprobable symbols, ordering the holes by their escape rate corresponds to neither the order by their measure nor by the length of the shortest periodic orbit they contain. Finally, we adapt our technique to the case of shifts endowed with Markov measures, where preliminary results show that a more intricate situation is to be expected.

math.DS↗

arXiv2022

On a waiting-time result of Kontoyiannis: mixing or decoupling?

We introduce conditions of lower decoupling to the study of waiting-time estimations of the cross entropy between two mutually independent stationary stochastic processes. Although similar decoupling conditions have been used in the literature on large deviations and statistical mechanics, they appear largely unexplored in information theory. Building on a result of Kontoyiannis, namely Theorem 4 in [Kontoyiannis, J. Theor. Probab., 1998], and replacing the $ψ$-mixing condition in this result with a lower decoupling condition, we considerably extend the validity of waiting-time estimation of cross entropy.

math.PR↗

arXiv2020

Transport properties and ageing for the averaged Lévy-Lorentz gas

We consider a persistent random walk on an inhomogeneous environment where the reflection probability depends only on the distance from the origin. Such an environment is the result of an average over all realizations of disorder of a Lévy-Lorentz (LL) gas. Here we show that this averaged Lévy-Lorentz gas yields nontrivial results even when the related LL gas is trivial. In particular, we investigate its long time transport properties such as the mean square displacement and the statistics of records, as well as the occurrence of ageing phenomena.

cond-mat.stat-mech↗

arXiv2020

Dynamics of transposable elements generates structure and symmetries in genetic sequences

Genetic sequences are known to possess non-trivial composition together with symmetries in the frequencies of their components. Recently, it has been shown that symmetry and structure are hierarchically intertwined in DNA, suggesting a common origin for both features. However, the mechanism leading to this relationship is unknown. Here we investigate a biologically motivated dynamics for the evolution of genetic sequences. We show that a metastable (long-lived) regime emerges in which sequences have symmetry and structure interlaced in a way that matches that of extant genomes.

q-bio.GN↗

arXiv2019

Natural Language Statistical Features of LSTM-generated Texts

Long Short-Term Memory (LSTM) networks have recently shown remarkable performance in several tasks dealing with natural language generation, such as image captioning or poetry composition. Yet, only few works have analyzed text generated by LSTMs in order to quantitatively evaluate to which extent such artificial texts resemble those generated by humans. We compared the statistical structure of LSTM-generated language to that of written natural language, and to those produced by Markov models of various orders. In particular, we characterized the statistical structure of language by assessing word-frequency statistics, long-range correlations, and entropy measures. Our main finding is that while both LSTM and Markov-generated texts can exhibit features similar to real ones in their word-frequency statistics and entropy measures, LSTM-texts are shown to reproduce long-range correlations at scales comparable to those found in natural language. Moreover, for LSTM networks a temperature-like parameter controlling the generation process shows an optimal value---for which the produced texts are closest to real language---consistent across all the different statistical features investigated.

cs.CL↗

arXiv2018

The common origin of symmetry and structure in genetic sequences

Biologists have long sought a way to explain how statistical properties of genetic sequences emerged and are maintained through evolution. On the one hand, non-random structures at different scales indicate a complex genome organisation. On the other hand, single-strand symmetry has been scrutinised using neutral models in which correlations are not considered or irrelevant, contrary to empirical evidence. Different studies investigated these two statistical features separately, reaching minimal consensus despite sustained efforts. Here we unravel previously unknown symmetries in genetic sequences, which are organized hierarchically through scales in which non-random structures are known to be present. These observations are confirmed through the statistical analysis of the human genome and explained through a simple domain model. These results suggest that domain models which account for the cumulative action of mobile elements can explain simultaneously non-random structures and symmetries in genetic sequences.

q-bio.QM↗

arXiv2018

Non-homogeneous persistent random walks and averaged environment for the Lévy-Lorentz gas

We consider transport properties for a non-homogeneous persistent random walk, that may be viewed as a mean-field version of the Lévy-Lorentz gas, namely a 1-d model characterized by a fat polynomial tail of the distribution of scatterers' distance, with parameter $α$. By varying the value of $α$ we have a transition from normal transport to superdiffusion, which we characterize by appropriate continuum limits.

cond-mat.stat-mech↗

arXiv2016

Random walks in a one-dimensional Lévy random environment

We consider a generalization of a one-dimensional stochastic process known in the physical literature as Lévy-Lorentz gas. The process describes the motion of a particle on the real line in the presence of a random array of marked points, whose nearest-neighbor distances are i.i.d. and long-tailed (with finite mean but possibly infinite variance). The motion is a continuous-time, constant-speed interpolation of a symmetric random walk on the marked points. We first study the quenched random walk on the point process, proving the CLT and the convergence of all the accordingly rescaled moments. Then we derive the quenched and annealed CLTs for the continuous-time process.

math.PR↗

arXiv2015

Characterization of DNA methylation as a function of biological complexity via dinucleotide inter-distances

We perform a statistical study of the distances between successive occurrencies of a given dinucleotide in the DNA sequence for a number of organisms of different complexity. Our analysis highlights peculiar features of the dinucleotide CG distribution in mammalian DNA, pointing towards a connection with the role of such dinucleotide in DNA methylation. While the CG distributions of mammals exhibit exponential tails with comparable parameters, the picture for the other organisms studied (e.g., fish, insects, bacteria and viruses) is more heterogeneous, possibly because in these organisms DNA methylation has different functional roles. Our analysis suggests that the distribution of the distances between dinucleotides CG provides useful insights in characterizing and classifying organisms in terms of methylation functionalities.

q-bio.QM↗

arXiv2015

Temporal-varying failures of nodes in networks

We consider networks in which random walkers are removed because of the failure of specific nodes. We interpret the rate of loss as a measure of the importance of nodes, a notion we denote as failure-centrality. We show that the degree of the node is not sufficient to determine this measure and that, in a first approximation, the shortest loops through the node have to be taken into account. We propose approximations of the failure-centrality which are valid for temporal-varying failures and we dwell on the possibility of externally changing the relative importance of nodes in a given network, by exploiting the interference between the loops of a node and the cycles of the temporal pattern of failures. In the limit of long failure cycles we show analytically that the escape in a node is larger than the one estimated from a stochastic failure with the same failure probability. We test our general formalism in two real-world networks (air-transportation and e-mail users) and show how communities lead to deviations from predictions for failures in hubs.

physics.soc-ph↗