arXiv subjects

Giambattista Giacomin

Publications and source records attributed to Giambattista Giacomin.

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↗

arXiv2026

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↗

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↗

arXiv2025

On the Lyapunov exponent for the random field Ising transfer matrix, in the critical case

We study the top Lyapunov exponent of a product of random $2 \times 2$ matrices appearing in the analysis of several statistical mechanical models with disorder, extending a previous treatment of the critical case (Giacomin and Greenblatt, ALEA 19 (2022), 701-728) by significantly weakening the assumptions on the disorder distribution. The argument we give completely revisits and improves the previous proof. As a key novelty we build a probability that is close to the Furstenberg probability, i.e. the invariant probability of the Markov chain corresponding to the evolution of the direction of a vector in ${\mathbb R}^2$ under the action of the random matrices, in terms of the ladder times of a centered random walk which is directly related to the random matrix sequence. We then show that sharp estimates on the ladder times (renewal) process lead to a sharp control on the probability measure we build and, in turn, to the control of its distance from the Furstenberg probability.

math.PR↗

arXiv2025

The random field Ising chain domain-wall structure in the large interaction limit

We study the configurations of the nearest neighbor Ising ferromagnetic chain with IID centered and square integrable external random field in the limit in which the pairwise interaction tends to infinity. The available free energy estimates for this model show a strong form of disorder relevance, i.e., a strong effect of disorder on the free energy behavior, and our aim is to make explicit how the disorder affects the spin configurations. We give a quantitative estimate that shows that the infinite volume spin configurations are close to one explicit disorder dependent configuration when the interaction is large. Our results confirm predictions on this model obtained in D. S. Fisher, P. Le Doussal and C. Monthus (Phys. Rev. E 2001) by applying the renormalization group method introduced by D. S. Fisher (Phys. Rev. B 1995).

math.PR↗

arXiv2024

Infinite disorder renormalization fixed point for the continuum random field Ising chain

We consider the continuum version of the random field Ising model in one dimension: this model arises naturally as weak disorder scaling limit of the original Ising model. Like for the Ising model, a spin configuration is conveniently described as a sequence of spin domains with alternating signs (domain-wall structure). We show that for fixed centered external field and as spin-spin couplings become large, the domain-wall structure scales to a disorder dependent limit that coincides with the infinite disorder fixed point process introduced by D. S. Fisher in the context of zero temperature quantum Ising chains. In particular, our results establish a number of predictions that one can find in [Fisher, Le Doussal and Monthus 2001]. The infinite disorder fixed point process for centered external field is equivalently described in terms of the process of suitably selected extrema of a Brownian trajectory introduced and studied by J. Neveu and J. Pitman [Neveu and Pitman 1989]. This characterization of the infinite disorder fixed point is one of the important ingredients of our analysis.

math.PR↗

arXiv2022

The zeros of the partition function of the pinning model

We aim at understanding for which (complex) values of the potential the pinning partition function vanishes. The pinning model is a Gibbs measure based on discrete renewal processes with power law inter-arrival distributions. We obtain some results for rather general inter-arrival laws, but we achieve a substantially more complete understanding for a specific one parameter family of inter-arrivals. We show, for such a specific family, that the zeros asymptotically lie on (and densely fill) a closed curve that, unsurprisingly, touches the real axis only in one point (the critical point of the model). We also perform a sharper analysis of the zeros close to the critical point and we exploit this analysis to approach the challenging problem of Griffiths singularities for the disordered pinning model. The techniques we exploit are both probabilistic and analytical. Regarding the first, a central role is played by limit theorems for heavy tail random variables. As for the second, potential theory and singularity analysis of generating functions, along with their interplay, will be at the heart of several of our arguments.

math-ph↗

arXiv2022

Lyapunov exponent for products of random Ising transfer matrices: the balanced disorder case

We analyze the top Lyapunov exponent of the product of sequences of two by two matrices that appears in the analysis of several statistical mechanics models with disorder: for example these matrices are the transfer matrices for the nearest neighbor Ising chain with random external field, and the free energy density of this Ising chain is the Lyapunov exponent we consider. We obtain the sharp behavior of this exponent in the large interaction limit when the external field is centered: this balanced case turns out to be critical in many respects. From a mathematical standpoint we precisely identify the behavior of the top Lyapunov exponent of a product of two dimensional random matrices close to a diagonal random matrix for which top and bottom Lyapunov exponents coincide. In particular, the Lyapunov exponent is only $\log$-Hölder continuous.

math.PR↗

arXiv2020

Localization, big-jump regime and the effect disorder for a class of generalized pinning models

One dimensional pinning models have been widely studied in the physical and mathematical literature, also in presence of disorder. Roughly speaking, they undergo a transition between a delocalized phase and a localized one. In mathematical terms these models are obtained by modifying the distribution of a discrete renewal process via a Boltzmann factor with an energy that contains only one body potentials. For some more complex models, notably pinning models based on higher dimensional renewals, it has been shown that other phases may be present. We study a generalization of the one dimensional pinning model in which the energy may depend in a nonlinear way on the contact fraction: this class of models contains the circular DNA case considered in the bio-physics literature. We give a full solution of this generalized pinning model in absence of disorder and show that another transition appears. In fact the systems may display up to three different regimes: delocalization, partial localization and full localization. What happens in the partially localized regime can be explained in terms of the "big-jump" phenomenon for sums of heavy tail random variables under conditioning. We then show that disorder completely smears this second transition and we are back to the delocalization versus localization scenario. In fact we show that the disorder, even if arbitrarily weak, is incompatible with the presence of a big-jump.

math-ph↗

arXiv2019

The disordered lattice free field pinning model approaching criticality

We continue the study, initiated in [Giacomin and Lacoin, JEMS 2018], of the localization transition of a lattice free field $ϕ=(ϕ(x))_{x \in Z^d}$, $d\ge 3$, in presence of a quenched disordered substrate. The presence of the substrate affects the interface at the spatial sites in which the interface height is close to zero. This corresponds to the Hamiltonian $$ \sum_{x\in Z^d }(βω_x+h)δ_x,$$ where $δ_x=1_{[-1,1]}(ϕ(x))$, and $(ω_x)_{x\in Z^d}$ is an IID centered field. A transition takes place when the average pinning potential $h$ goes past a threshold $h_c(β)$: from a delocalized phase $h h_c(β)$ where the field sticks to the substrate. In [Giacomin and Lacoin, JEMS 2018] the critical value of $h$ is identified and it coincides, up to the sign, with the $\log$-Laplace transform of $ω=ω_x$, that is $-h_c(β)=λ(β):=\log E[e^{βω}]$. Here we obtain the sharp critical behavior of the free energy approaching criticality: $$\lim_{u\searrow 0} \frac{ F(β,h_c(β)+u)}{u^2}= \frac{1}{2\, \textrm{Var}\left(e^{βω-λ(β)}\right)}.$$ Moreover, we give a precise description of the trajectories of the field in the same regime: the absolute value of the field is $\sqrt{2σ_d^2\vert\log(h-h_c(β))\vert}$ to leading order when $h\searrow h_c(β)$ except on a vanishing fraction of sites ($σ_d^2$ is the single site variance of the free field).

math-ph↗

arXiv2019

A Law of Large Numbers and Large Deviations for interacting diffusions on Erdős-Rényi graphs

We consider a class of particle systems described by differential equations (both stochastic and deterministic), in which the interaction network is determined by the realization of an Erdős-Rényi graph with parameter $p_n\in (0, 1]$, where $n$ is the size of the graph (i.e., the number of particles). If $p_n\equiv 1$ the graph is the complete graph (mean field model) and it is well known that, under suitable hypotheses, the empirical measure converges as $n\to \infty$ to the solution of a PDE: a McKean-Vlasov (or Fokker-Planck) equation in the stochastic case, or a Vlasov equation in the deterministic one. It has already been shown that this holds for rather general interaction networks, that include Erdős-Rényi graphs with $\lim_n p_n n =\infty$, and properly rescaling the interaction to account for the dilution introduced by $p_n$. However, these results have been proven under strong assumptions on that initial datum which has to be chaotic, i.e. a sequence of independent identically distributed random variables. The aim of our contribution is to present results -- Law of Large Numbers and Large Deviation Principle -- assuming only the convergence of the empirical measure of the initial condition.

math.PR↗

arXiv2018

Continuum limit of random matrix products in statistical mechanics of disordered systems

We consider a particular weak disorder limit ("continuum limit") of matrix products that arise in the analysis of disordered statistical mechanics systems, with a particular focus on random transfer matrices. The limit system is a diffusion model for which the leading Lyapunov exponent can be expressed explicitly in terms of modified Bessel functions, a formula that appears in the physical literature on these disordered systems. We provide an analysis of the diffusion system as well as of the link with the matrix products. We then apply the results to the framework considered by Derrida and Hilhorst [J. Phys. A (1983)], which deals in particular with the strong interaction limit for disordered Ising model in one dimension and that identifies a singular behavior of the Lyapunov exponent (of the transfer matrix), and to the two dimensional Ising model with columnar disorder (McCoy-Wu model). We show that the continuum limit sharply captures the Derrida and Hilhorst singularity. Moreover we revisit the analysis by McCoy and Wu [Phys. Rev. 1968] and remark that it can be interpreted in terms of the continuum limit approximation. We provide a mathematical analysis of the continuum approximation of the free energy of the McCoy-Wu model, clarifying the prediction (by McCoy and Wu) that, in this approximation, the free energy of the two dimensional Ising model with columnar disorder is $C^\infty$ but not analytic at the critical temperature.

math-ph↗

arXiv2018

Disorder and denaturation transition in the generalized Poland-Scheraga model

We investigate the generalized Poland-Scheraga model, which is used in the bio-physical literature to model the DNA denaturation transition, in the case where the two strands are allowed to be non-complementary (and to have different lengths). The homogeneous model was recently studied from a mathematical point of view in Giacomin, Khatib (Stoch. Proc. Appl., 2017), via a $2$-dimensional renewal approach, with a loop exponent $2+α$ (${α>0}$): it was found to undergo a localization/delocalization phase transition of order $ν= \min(1,α)^{-1}$, together with -- in general -- other phase transitions. In this paper, we turn to the disordered model, and we address the question of the influence of disorder on the denaturation phase transition, that is whether adding an arbitrarily small amount of disorder (i.e. inhomogeneities) affects the critical properties of this transition. Our results are consistent with Harris' predictions for $d$-dimensional disordered systems (here $d=2$). First, we prove that when $α<1$ (i.e. $ν>d/2$), then disorder is irrelevant: the quenched and annealed critical points are equal, and the disordered denaturation phase transition is also of order $ν=α^{-1}$. On the other hand, when $α>1$, disorder is relevant: we prove that the quenched and annealed critical points differ. Moreover, we discuss a number of open problems, in particular the smoothing phenomenon that is expected to enter the game when disorder is relevant.

math.PR↗

arXiv2017

Disorder and critical phenomena: the $α=0$ copolymer model

The generalized copolymer model is a disordered system built on a discrete renewal process with inter-arrival distribution that decays in a regularly varying fashion with exponent $1+ α\geq 1$. It exhibits a localization transition which can be characterized in terms of the free energy of the model: the free energy is zero in the delocalized phase and it is positive in the localized phase. This transition, which is observed when tuning the mean $h$ of the disorder variable, has been tackled in the physics literature notably via a renormalization group procedure that goes under the name of \emph{strong disorder renormalization}. We focus on the case $α=0$ -- the critical value $h_c(β)$ of the parameter $h$ is exactly known (for every strength $β$ of the disorder) in this case -- and we provide precise estimates on the critical behavior. Our results confirm the strong disorder renormalization group prediction that the transition is of infinite order, namely that when $h\searrow h_c(β)$ the free energy vanishes faster than any power of $h-h_c(β)$. But we show that the free energy vanishes much faster than the physicists' prediction.

math.PR↗

arXiv2017

DNA melting structures in the generalized Poland-Scheraga model

The Poland-Scheraga model for DNA denaturation, besides playing a central role in applications, has been widely studied in the physical and mathematical literature over the past decades. More recently a natural generalization has been introduced in the biophysics literature to overcome the limits of the original model, namely to allow an excess of bases -- i.e. a different length of the two single stranded DNA chains -- and to allow slippages in the chain pairing. The increased complexity of the model is reflected in the appearance of configurational transitions when the DNA is in double stranded form. In a previous work of two of the authors the generalized Poland-Scheraga model has been analyzed thanks to a representation in terms of a bivariate renewal process. In this work we exploit this representation farther and fully characterize the path properties of the system, making therefore explicit the geometric structures -- and the configurational transitions -- that are observed when the polymer is in the double stranded form. What we prove is that, when the excess of bases is not absorbed in a homogeneous fashion along the double stranded chain, then it either condensates in a single macroscopic loop or it accumulates into an unbound single strand free end.

math.PR↗

arXiv2016

Singular behavior of the leading Lyapunov exponent of a product of random $2 \times 2$ matrices

We consider a certain infinite product of random $2 \times 2$ matrices appearing in the solution of some $1$ and $1+1$ dimensional disordered models in statistical mechanics, which depends on a parameter $\varepsilon>0$ and on a real random variable with distribution $μ$. For a large class of $μ$, we prove the prediction by B. Derrida and H. J. Hilhorst (J. Phys. A 16:2641, 1983) that the Lyapunov exponent behaves like $C \varepsilon^{2 α}$ in the limit $\varepsilon \searrow 0$, where $α\in (0,1)$ and $C>0$ are determined by $μ$. Derrida and Hilhorst performed a two-scale analysis of the integral equation for the invariant distribution of the Markov chain associated to the matrix product and obtained a probability measure that is expected to be close to the invariant one for small $\varepsilon$. We introduce suitable norms and exploit contractivity properties to show that such a probability measure is indeed close to the invariant one in a sense which implies a suitable control of the Lyapunov exponent.

math-ph↗

arXiv2016

A note on dynamical models on random graphs and Fokker-Planck equations

We address the issue of the proximity of interacting diffusion models on large graphs with a uniform degree property and a corresponding mean field model, i.e. a model on the complete graph with a suitably renormalized interaction parameter. Examples include Erdős-Rényi graphs with edge probability $p_n$, $n$ is the number of vertices, such that $\lim_{n \to \infty}p_n n= \infty$. The purpose of this note it twofold: (1) to establish this proximity on finite time horizon, by exploiting the fact that both systems are accurately described by a Fokker-Planck PDE (or, equivalently, by a nonlinear diffusion process) in the $n=\infty$ limit; (2) to remark that in reality this result is unsatisfactory when it comes to applying it to systems with $N$ large but finite, for example the values of $N$ that can be reached in simulations or that correspond to the typical number of interacting units in a biological system.

math.PR↗

arXiv2016

Disorder and wetting transition: the pinned harmonic crystal in dimension three or larger

We consider the Lattice Gaussian free field in $d+1$ dimensions, $d=3$ or larger, on a large box (linear size $N$) with boundary conditions zero. On this field two potentials are acting: one, that models the presence of a wall, penalizes the field when it enters the lower half space and one, the «pinning potential» , that rewards visits to the proximity of the wall. The wall can be soft, i.e. the field has a finite penalty to enter the lower half plane, or hard when the penalty is infinite. In general the pinning potential is disordered and it gives on average a reward h in $\mathbb{R}$ (a negative reward is a penalty): the energetic contribution when the field at site x visits the pinning region is $βω_x+h$, $\{ω_x\}_{x \in \mathbb{Z}^d}$ are IID centered and exponentially integrable random variables of unit variance and $β\ge 0$. In [E. Bolthausen, J.-D. Deuschel and O. Zeitouni, J. Math. Phys. 41 (2000), 1211-1223] it is shown that, when $β=0$ (that is, in the non disordered model), a delocalization-localization transition happens at $h=0$, in particular the free energy of the system is zero for $h \le 0$ and positive for $h>0$. We show that, for $β\neq 0$, the transition happens at $h=h_c(β):=- \log \mathbb{E} \exp(βω_x)$ and we find the precise asymptotic behavior of the logarithm of the free energy density of the system when $h \searrow h_c(β)$. In particular, we show that the transition is of infinite order in the sense that the free energy is smaller than any power of $h-h_c(β)$ in the neighborhood of the critical point and that disorder does not modify at all the nature of the transition. We also provide results on the behavior of the paths of the random field in the limit $N \to \infty$.

math-ph↗