arXiv subjects

Giuseppe Scola

Publications and source records attributed to Giuseppe Scola.

7 recordsLinked to original sources

arXiv2026

New convergence bound for the cluster expansion in canonical ensemble

We perform a cluster expansion in the canonical ensemble with periodic boundary conditions, introducing a new choice of polymer activities that differs from the standard ones. This choice leads to an improved bound for the convergence of the cluster expansion, which we compare with the known one. We also recover the irreducible Mayer coefficients for the thermodynamic free energy. The results presented here can also be applied to the case of zero boundary conditions and to the convergence of correlation expansions.

math-ph↗

arXiv2026

The consensus problem for opinion dynamics with local average random interactions

We study the consensus formation for an agents based model, generalizing that originally proposed by Krause \cite{Kr}, by allowing the communication channels between any couple of agents to be switched on or off randomly, at each time step, with a probability law depending on the proximity of the agents' opinions. Namely, we consider a system of agents sharing their opinions according to the following updating protocol. At time $t+1$ the opinion $X_{i}\left( t+1\right) \in\left[ 0,1\right] $ of any agent $i$ is updated at the weighted average of the opinions of the agents communicating with it at time $t.$ The weights model the confidence level an agent assigns to the opinions of the other agents and are kept fixed by the system dynamics, but the set of agents communicating with any agent $i$ at time $t+1$ is randomly updated in such a way that the agent $j$ can be chosen to belong to this set independently of the other agents with a probability that is a non increasing function of $\left\vert X_{i}\left( t\right) -X_{j}\left(t\right) \right\vert .$ This condition models the fact that a communication among the agents is more likely to happen if their opinions are close. We prove that if the agent's communication graph at time one, conditionally on the initial believes' configuration, is sufficiently connected, the system reaches consensus at geometric rate, i.e., more precisely, as the time tends to infinity the agents' opinions will reach the same value geometrically fast. We also discuss the consensus formation for a system of infinitely many agents. In particular we analyze the evolution of the empirical average of the agents' opinions in the limit as the size of the system tends to infinity and characterize its fixed points in terms of agents' consensus proving that this is reached geometrically fast with the same rate computed for the finite system.

cs.SI↗

arXiv2025

Non-trivial fixed point of a $ψ^4_d$ fermionic theory, II. Anomalous exponent and scaling operators

We consider the Renormalization Group (RG) fixed-point theory associated with a fermionic $ψ^4_d$ model in $d=1,2,3$ with fractional kinetic term, whose scaling dimension is fixed so that the quartic interaction is weakly relevant in the RG sense. The model is defined in terms of a Grassmann functional integral with interaction $V^*$, solving a fixed-point RG equation in the presence of external fields, and a fixed ultraviolet cutoff. We define and construct the field and density scale-invariant response functions, and prove that the critical exponent of the former is the naive one, while that of the latter is anomalous and analytic. We construct the corresponding (almost-)scaling operators, whose two point correlations are scale-invariant up to a remainder term, which decays like a stretched exponential at distances larger than the inverse of the ultraviolet cutoff. Our proof is based on constructive RG methods and, specifically, on a convergent tree expansion for the generating function of correlations, which generalizes the approach developed by three of the authors in a previous publication [A. Giuliani, V. Mastropietro, S. Rychkov, JHEP 01 (2021) 026].

math-ph↗

arXiv2025

Large scale dynamical response of interacting $1d$ Fermi systems

We consider the dynamics of a class of weakly interacting, gapless $1d$ fermionic systems, in presence of small external perturbations slowly varying in space and in time. We consider the evolution of the expectation values of the charge density and of the current density, in the thermodynamic limit and for low enough temperatures. We prove the validity and the asymptotic exactness of linear response in the limit of vanishing space-time variation of the perturbation, and we provide the explicit expression of the response of the system. The proof relies on the representation of the real time Duhamel expansion in terms of Euclidean correlation functions, for which we provide sharp estimates using rigorous renormalization group methods. The asymptotic exactness of linear response holds thanks to a cancellation for the scaling limit of the correlations that is reminiscent of bosonization, and which is derived rigorously using emergent chiral Ward identities.

math-ph↗

arXiv2023

Free energy expansions for renormalized systems for colloids

We consider a binary system of small and large spheres of finite size in a continuous medium interacting via a non-negative potential. We work in the canonical ensemble and compute upper and lower bound for the free energy at finite and infinite volume by first integrating over the small spheres and then treating the effective system of the large ones which now interact via a multi-body potential. By exploiting the underlying structure of the original binary system we prove the convergence of the cluster expansion for the latter system and obtain a sufficient condition which involves the surface of the large spheres rather than their volume (as it would have been the case in a direct application of existing methods directly to the binary system). Our result is valid for the particular case of hard spheres (colloids) for which we rigorously treat the depletion interaction.

math-ph↗

arXiv2021

Cluster expansion for the Ising model in the canonical ensemble

We show the validity of the cluster expansion in the canonical ensemble for the Ising model. We compare the lower bound of its radius of convergence with the one computed by the virial expansion working in the grand-canonical ensemble. Using the cluster expansion we give direct proofs with quantification of the higher order error terms for the decay of correlations, central limit theorem and large deviations.

math-ph↗

arXiv2020

Local moderate and precise large deviations via cluster expansions

We consider a system of classical particles confined in a box $Λ\subset\mathbb{R}^d$ with zero boundary conditions interacting via a stable and regular pair potential. Based on the validity of the cluster expansion for the canonical partition function in the high temperature - low density regime we prove moderate and precise large deviations from the mean value of the number of particles with respect to the grand-canonical Gibbs measure. In this way we have a direct method of computing both the exponential rate as well as the pre-factor and obtain explicit error terms. Estimates comparing with the infinite volume versions of the above are also provided.

math.PR↗