arXiv subjects

Marco Zamparo

Publications and source records attributed to Marco Zamparo.

At least 19 recordsLinked to original sources

arXiv2026

The localized phase of pinning models with correlated Gaussian disorder

We demonstrate that the results established for the localized regime of the pinning model with independent disorder -- notably the $\mathcal{C}^\infty$ regularity of the free energy, the scaling of the largest excursion between pinned sites, and the Central Limit Theorems for both the contact number and its mean -- can be generalized to translation-ergodic environments under the hypothesis that disorder is Gaussian. Our results are obtained assuming only summability of the charges' covariances. The two key ingredients for our proofs are the Birkhoff-sum approach introduced in Giacomin, Zamparo (2024) for independent disorder, but uniquely suited to handle correlated environments, and decorrelation tools such as the general and powerful Nelson's Gaussian hypercontractivity. We also rely on the Bernstein blocking method to prove a Central Limit Theorem under dependence. Additionally, we develop further techniques specifically tailored to the one-dimensional structure of the pinning model.

math.PR↗

arXiv2025

Pinning models with contact number constraint: the effect of disorder

Disordered pinning models are statistical mechanics models built on discrete renewal processes: renewal epochs in this context are called contacts. It is well known that pinning models can undergo a localization/delocalization phase transition: in the localized phase the typical density of contacts is positive and the largest gap between contacts is at most of the order of the logarithm of the size $n$ of the system, whereas the system is void of contacts in the delocalized phase. When disorder is absent and the phase transition is discontinuous, conditioning the contact density to be positive but smaller than the minimum typical density in the localized phase has the effect of forcing to create one, and only one, macroscopic gap between two contacts, while the rest of the configuration keeps the characteristics of a localized state. However it is known that, in the presence of (bounded) disorder, this big jump phenomenon is no longer observed, in the sense that the largest gap in the conditioned system is $o(n)$. Under minimal integrability conditions on the disorder we show that the conditioned system is localized in a very strong sense, in particular the largest gap is $O(\log n)$. The proof is achieved by exploiting the improved understanding of the localized phase of disordered pinning models developed in [34] and by establishing some refined estimates, in particular a quenched Local Central Limit Theorem. We also present an analysis of the effect of disorder on the large deviation rate function of the contact density and, as an important ingredient for the generality of our results, we establish a smoothing inequality for pinning models under minimal integrability conditions, thus generalizing [11,32].

math.PR↗

arXiv2024

Work fluctuations for a confined Brownian particle: the role of initial conditions

We study the large fluctuations of the work injected by the random force into a Brownian particle under the action of a confining harmonic potential. In particular, we compute analytically the rate function for generic uncorrelated initial conditions, showing that, depending on the initial spread, it can exhibit no, one, or two singularities associated to the onset of linear tails. A dependence on the potential strength is observed for large initial spreads (entailing two singularities), which is lost for stationary initial conditions (giving one singularity) and concentrated initial values (no singularity). We discuss the mechanism responsible for the singularities of the rate function, identifying it as a big jump in the initial values. Analytical results are corroborated by numerical simulations.

cond-mat.stat-mech↗

arXiv2024

Concentration and fluctuation phenomena in the localized phase of the pinning model

We focus on the localized phase of pinning models with i.i.d. site disorder on which we assume only that the moment generating function is bounded in a neighborhood of the origin. We develop quantitative correlation functions estimates for local observables that entail quantitative $C^\infty$ estimates on the free energy density, showing in particular that its regularity class is at least Gevrey-3 in the whole localized phase. We then explain how a quenched concentration bound and the quenched Central Limit Theorem (CLT) on the number of the pinned sites, i.e., the $contact~number$, can be extracted from the regularity estimates on the free energy: this identifies the thermal fluctuations of the contact number. But the centering sequence in the quenched CLT is random in the sense that it is disorder dependent: we show that the (disorder induced) fluctuations of the centering are on the same scale of the thermal fluctuations by establishing a CLT, with a non degenerate variance, also for the centering. For what concerns the correlation and $C^\infty$ estimates, our work substantially generalizes and expands the analysis in [Giacomin and F. L. Toninelli, Lat. Am. J. Probab. 1 (2006), 149-180] that dealt with pinning models with restrictive conditions on the disorder distributions and in which less explicit, non uniform bounds were obtained.

math.PR↗

arXiv2023

Quenched large deviations in renewal theory

In this paper we introduce and study renewal-reward processes in random environments where each renewal involves a reward taking values in a Banach space. We derive quenched large deviation principles and identify the associated rate functions in terms of variational formulas involving correctors. We illustrate the theory with three examples: compound Poisson processes in random environments, pinning of polymers at interfaces with disorder, and returns of Markov chains in dynamic random environments.

math.PR↗

arXiv2023

Work fluctuations for a harmonically confined Active Ornstein-Uhlenbeck Particle

We study the active work fluctuations of an active Ornstein-Uhlenbeck particle in the presence of a confining harmonic potential. We tackle the problem analytically both for stationary and generic uncorrelated initial states. Our results show that harmonic confinement can induce singularities in the active work rate function, with linear stretches at large positive and negative active work, at sufficiently large active and harmonic force constants. These singularities originate from big jumps in the displacement and in the active force, occurring at the initial or ending points of trajectories and marking the relevance of boundary terms in this problem.

cond-mat.stat-mech↗

arXiv2022

Statistical fluctuations under resetting: rigorous results

In this paper we investigate the normal and the large fluctuations of additive functionals associated with a stochastic process under a general non-Poissonian resetting mechanism. Cumulative functionals of regenerative processes are very close to renewal-reward processes and inherit most of the properties of the latter. Here we review and use the classical law of large numbers and central limit theorem for renewal-reward processes to obtain same theorems for additive functionals of a stochastic process under resetting. Then, we establish large deviation principles for these functionals by illustrating and applying a large deviation theory for renewal-reward processes that has been recently developed by the author. We discuss applications of the general results to the positive occupation time, the area, and the absolute area of the reset Brownian motion. While introducing advanced tools from renewal theory, we demonstrate that a rich phenomenology accounting for dynamical phase transitions emerges when one goes beyond Poissonian resetting.

math.PR↗

arXiv2022

Large fluctuations and transport properties of the Lévy-Lorentz gas

The Lévy-Lorentz gas describes the motion of a particle on the real line in the presence of a random array of scattering points, whose distances between neighboring points are heavy-tailed i.i.d. random variables with finite mean. The motion is a continuous-time, constant-speed interpolation of the simple symmetric random walk on the marked points. In this paper we study the large fluctuations of the continuous-time process and the resulting transport properties of the model, both annealed and quenched, confirming and extending previous work by physicists that pertain to the annealed framework. Specifically, focusing on the particle displacement, and under the assumption that the tail distribution of the interdistances between scatterers is regularly varying at infinity, we prove a precise large deviation principle for the annealed fluctuations and present the asymptotics of annealed moments, demonstrating annealed superdiffusion. Then, we provide an upper large deviation estimate for the quenched fluctuations and the asymptotics of quenched moments, showing that the asymptotic diffusive regime conditional on a typical arrangement of the scatterers is normal diffusion, and not superdiffusion. Although the Lévy-Lorentz gas seems to be accepted as a model for anomalous diffusion, our findings suggest that superdiffusion is a transient behavior which develops into normal diffusion on long timescales, and raise a new question about how the transition from the quenched normal diffusion to the annealed superdiffusion occurs.

math.PR↗

arXiv2022

Phase separation and critical size in molecular sorting

Molecular sorting is a fundamental process that allows eukaryotic cells to distill and concentrate specific chemical factors in appropriate cell membrane subregions, thus endowing them with different chemical identities and functional properties. A phenomenological theory of this molecular distillation process has recently been proposed [arXiv:1811.06760], based on the idea that molecular sorting emerges from the combination of: a) phase-separation-driven formation of sorting domains, and b) domain-induced membrane bending, leading to the production of submicrometric lipid vesicles enriched in the sorted molecules. In this framework, a natural parameter controlling the efficiency of molecular distillation is the critical size of phase-separated domains. In the experiments, sorting domains appear to fall into two classes: unproductive domains, characterized by short lifetimes and low probability of extraction, and productive domains, that evolve into vesicles that ultimately detach from the membrane system. It is tempting to link these two classes to the different fates predicted by classical phase separation theory for subcritical and supercritical phase-separated domains. Here, we discuss the implication of this picture in the framework of the previously introduced phenomenological theory of molecular sorting. Several predictions of the theory are verified by numerical simulations of a lattice-gas model. Sorting is observed to be most efficient when the number of sorting domains is close to a minimum. To help in the analysis of experimental data, an operational definition of the critical size of sorting domains is proposed. Comparison with experimental results shows that the statistical properties of productive/unproductive domains inferred from experimental data are in agreement with those predicted from numerical simulations of the model.

cond-mat.soft↗

arXiv2022

Large deviations for quadratic functionals of stable Gauss-Markov chains and entropy production

In this paper we establish a large deviation principle for the entropy production rate of possible non-stationary, centered stable Gauss-Markov chains, verifying the Gallavotti-Cohen symmetry. We reach this goal by developing a large deviation theory for quasi-Toeplitz quadratic functionals of multivariate centered stable Gauss-Markov chains, which differ from a perfect Toeplitz form by the addition of quadratic boundary terms.

math.PR↗

arXiv2022

Renewal model for dependent binary sequences

We suggest to construct infinite stochastic binary sequences by associating one of the two symbols of the sequence with the renewal times of an underlying renewal process. Focusing on stationary binary sequences corresponding to delayed renewal processes, we investigate correlations and the ability of the model to implement a prescribed autocovariance structure, showing that a large variety of subexponential decay of correlations can be accounted for. In particular, robustness and efficiency of the method are tested by generating binary sequences with polynomial and stretched-exponential decay of correlations. Moreover, to justify the maximum entropy principle for model selection, an asymptotic equipartition property for typical sequences that naturally leads to the Shannon entropy of the waiting time distribution is demonstrated. To support the comparison of the theory with data, a law of large numbers and a central limit theorem are established for the time average of general observables.

math-ph↗

arXiv2021

Large deviation principles for renewal-reward processes

We establish a sharp large deviation principle for renewal-reward processes, supposing that each renewal involves a broad-sense reward taking values in a real separable Banach space. In fact, we demonstrate a weak large deviation principle without assuming any exponential moment condition on the law of waiting times and rewards by resorting to a sharp version of Cramér's theorem. We also exhibit sufficient conditions for exponential tightness of renewal-reward processes, which leads to a full large deviation principle.

math.PR↗

arXiv2021

Large Deviations in Discrete-Time Renewal Theory

We establish sharp large deviation principles for cumulative rewards associated with a discrete-time renewal model, supposing that each renewal involves a broad-sense reward taking values in a real separable Banach space. The framework we consider is the pinning model of polymers, which amounts to a Gibbs change of measure of a classical renewal process and includes it as a special case. We first tackle the problem in a constrained pinning model, where one of the renewals occurs at a given time, by an argument based on convexity and super-additivity. We then transfer the results to the original pinning model by resorting to conditioning.

math.PR↗

arXiv2021

Critical Fluctuations in Renewal Models of Statistical Mechanics

We investigate the sharp asymptotic behavior at criticality of the large fluctuations of extensive observables in renewal models of statistical mechanics, such as the Poland-Scheraga model of DNA denaturation, the Fisher-Felderhof model of fluids, the Wako-Saitô-Muñoz-Eaton model of protein folding, and the Tokar-Dreyssé model of strained epitaxy. These models amount to Gibbs changes of measure of a classical renewal process and can be identified with a constrained pinning model of polymers. The extensive observables that enter the thermodynamic description turn out to be cumulative rewards corresponding to deterministic rewards that are uniquely determined by the waiting time and grow no faster than it. The probability decay with the system size of their fluctuations switches from exponential to subexponential at criticality, which is a regime corresponding to a discontinuous pinning-depinning phase transition. We describe such decay by proposing a precise large deviation principle under the assumption that the subexponential correction term to the waiting time distribution is regularly varying. This principle is in particular used to characterize the fluctuations of the number of renewals, which measures the DNA-bound monomers in the Poland-Scheraga model, the particles in the Fisher-Felderhof model and the Tokar-Dreyssé model, and the native peptide bonds in the Wako-Saitô-Muñoz-Eaton model.

math.PR↗

arXiv2021

Optimality in self-organized molecular sorting

We introduce a simple physical picture to explain the process of molecular sorting, whereby specific proteins are concentrated and distilled into submicrometric lipid vesicles in eukaryotic cells. To this purpose, we formulate a model based on the coupling of spontaneous molecular aggregation with vesicle nucleation. Its implications are studied by means of a phenomenological theory describing the diffusion of molecules towards multiple sorting centers that grow due to molecule absorption and are extracted when they reach a sufficiently large size. The predictions of the theory are compared with numerical simulations of a lattice-gas realization of the model and with experimental observations. The efficiency of the distillation process is found to be optimal for intermediate aggregation rates, where the density of sorted molecules is minimal and the process obeys simple scaling laws. Quantitative measures of endocytic sorting performed in primary endothelial cells are compatible with the hypothesis that these optimal conditions are realized in living cells.

cond-mat.stat-mech↗

arXiv2019

Large Deviations in Renewal Models of Statistical Mechanics

In Ref. [1] the author has recently established sharp large deviation principles for cumulative rewards associated with a discrete-time renewal model, supposing that each renewal involves a broad-sense reward taking values in a separable Banach space. The renewal model has been there identified with constrained and non-constrained pinning models of polymers, which amount to Gibbs changes of measure of a classical renewal process. In this paper we show that the constrained pinning model is the common mathematical structure to the Poland-Scheraga model of DNA denaturation and to some relevant one-dimensional lattice models of Statistical Mechanics, such as the Fisher-Felderhof model of fluids, the Wako-Saitô-Muñoz-Eaton model of protein folding, and the Tokar-Dreyssé model of strained epitaxy. Then, in the framework of the constrained pinning model, we develop an analytical characterization of the large deviation principles for cumulative rewards corresponding to multivariate deterministic rewards that are uniquely determined by, and at most of the order of magnitude of, the time elapsed between consecutive renewals. In particular, we outline the explicit calculation of the rate functions and successively we identify the conditions that prevent them from being analytic and that underlie affine stretches in their graphs. Finally, we apply the general theory to the number of renewals. From the point of view of Equilibrium Statistical Physics and Statistical Mechanics, cumulative rewards of the above type are the extensive observables that enter the thermodynamic description of the system. The number of renewals, which turns out to be the commonly adopted order parameter for the Poland-Scheraga model and for also the renewal models of Statistical Mechanics, is one of these observables.

math-ph↗

arXiv2019

Large Deviations in Renewal Theory and Renewal Models of Statistical Mechanics

We present and establish large deviations principles for general multivariate renewal-reward processes associated with a classical discrete-time renewal process. A renewal-reward process describes a cumulative reward over time, supposing that a broad-sense multivariate reward is obtained at each occurrence of the event that is renewed under the renewal process. We consider both the standard model and a constrained model that is constructed conditioning on the event that one of the renewals occurs at a predetermined time. With a different interpretation of the time coordinate, the constrained renewal model includes several important models of statistical mechanics, such as the model of polymer pinning, the Poland-Scheraga model of DNA denaturation, the Wako-Saitô-Muñoz-Eaton model of protein folding, and the Tokar-Dreyssé model of strained epitaxy. We attack the problem of large deviations in constrained renewal models by an argument based on convexity and super-additivity. Then, we transfer results to standard renewal processes by resorting to conditioning. In the context of constrained renewal models, we also propose an explicit application of the general theory to deterministic rewards that grow no faster than the time elapsed between two successive occurrences of the renewed event. This type of rewards codifies the extensive variables of statistical mechanics.

math-ph↗

arXiv2018

Dynamical transition in the TASEP with Langmuir kinetics: mean-field theory

We develop a mean-field theory for the totally asymmetric simple exclusion process (TASEP) with open boundaries, in order to investigate the so-called dynamical transition. The latter phenomenon appears as a singularity in the relaxation rate of the system toward its non-equilibrium steady state. In the high-density (low-density) phase, the relaxation rate becomes independent of the injection (extraction) rate, at a certain critical value of the parameter itself, and this transition is not accompanied by any qualitative change in the steady-state behavior. We characterize the relaxation rate by providing rigorous bounds, which become tight in the thermodynamic limit. These results are generalized to the TASEP with Langmuir kinetics, where particles can also bind to empty sites or unbind from occupied ones, in the symmetric case of equal binding/unbinding rates. The theory predicts a dynamical transition to occur in this case as well.

cond-mat.stat-mech↗