arXiv subjects

Simone Franchini

Publications and source records attributed to Simone Franchini.

At least 19 recordsLinked to original sources

arXiv2026

REM universality and Poisson-Dirichlet Gibbs weights for linear random energy

We study the Hamiltonian $H_n(h,\sigma)=\sum_{i=1}^n h_i(\sigma_i-m), $ where $(h_i)$ are i.i.d.\ real random variables and $(\sigma_i)$ are i.i.d.\ Ising spins. We consider the energy levels obtained after an independent thinning that retains an exponential number of configurations ($e^{O(n)}$). We prove that, after an $(h_i)$-dependent centering, the resulting point process converges in distribution to a Poisson point process with exponential intensity. Thus, the energy levels asymptotically has the one of the Random Energy Model (REM). Our results extend previous ones, where REM universality for this model was established only either for energy fluctuations of order $e^{-O(n)}$ or for $e^{o(\sqrt n)}$ randomly selected configurations. We also identify the limiting Gibbs weights, which converge to a Poisson--Dirichlet law, and the quenched free energy, which exhibits a freezing transition at $\beta=\tilde\lambda$. The proofs are presented here in compressed form; full details are given in the companion preprint.

math.PR↗

arXiv2026

REM universality for linear random energy

We consider a sequence of random Hamiltonians $H_n(h,\sigma)=\sum^n_{i=1}h_i(\sigma_i-m)$, and study the asymptotic ($n\to \infty$) distribution of the energy levels $(H_n(h,\sigma))_{\sigma\in \{-1,1\}^n}$, where $h_1,h_2,\cdots$ are i.i.d. random variables. We show that, when $e^{O(n)}$ configurations are sampled at random, the corresponding collection of energy levels converges in distribution to a Poisson point process with exponential intensity measure. This establishes the Random Energy Model (REM) universality for the present model. Our results strengthen earlier works on local REM universality by characterizing the distribution of $O(1)-$order fluctuations of $H_n$. In addition, we improve upon the REM universality by dilution studied by Ben Arous, Gayrard, Kuptsov by allowing an exponentially large number $e^{O(n)}$ of sampled configurations, instead of $e^{o(\sqrt{n})}$. Finally, we derive the asymptotic distribution of the Gibbs weight.

math.PR↗

arXiv2026

Lattice Field Theory for a network of real neurons

In a recent paper [Bardella et al., Entropy 26 (6), 495 (2024)] we introduced a simplified Lattice Field Theory (LFT) framework that allows experimental recordings from major Brain-Computer Interfaces (BCIs) to be interpreted in a simple and physically grounded way. From a neuroscience point of view, our method modifies the Maximum Entropy model for neural networks so that also the time evolution of the system is taken into account and it can be interpreted as another version of the Free Energy principle (FEP). This framework is naturally tailored to interpret recordings from chronic multi-site BCIs, especially spike rasters from measurements of single neuron activity.

cond-mat.stat-mech↗

arXiv2026

Constructive Cavity Method

We show that the functional appearing in the celebrated Parisi formula for the free energy of the Sherrington-Kirkpatrick model can be found from the incremental free energy obtained by Cavity Method if one assumes that the state is a product of independent Random Energy models.

cond-mat.stat-mech↗

arXiv2025

On the Pure States of the Replica Symmetry Breaking ansatz

We discuss the concept of Pure State of the Replica Symmetry Breaking ansatz in finite and infinite spin systems without averaging on the disorder, nor using replicas. Consider a system of n spins $\sigma\in\Omega^{n}$ with the usual set $\Omega=\left\{ -1,1\right\}$ of inner states and let $G:\,\Omega^{n}\rightarrow\left[0,1\right]$ a Gibbs measure on it of Hamiltonian $\mathcal{H}$ (also non random). We interpret the pure states of a model $\left(\Omega^{n},\mu\right)$ as disjoint subsets $\Omega^{n}$ such that the conditional measures behaves like product measures as in usual mean field approximations. Starting from such definition we try to reinterpret the RSB scheme and define an approximated probability measure. We then apply our results to the Sherrington-Kirkpatrick model to obtain the Parisi formula.

cond-mat.stat-mech↗

arXiv2025

Elephant Random Walk with multiple extractions

Consider a generalized Elephant Random Walk in which the step is chosen by selecting $k$ previous steps with $k$ odd and then going in the majority direction with a probability $p$ and in the opposite direction otherwise. In the $k=1$ case the model is the original one and could be resolved exactly by analogy with Friedman's urn. However the analogy cannot be extended to the $k>2$ case already. In this paper we show how to treat the model for each $k$ by analogy with the more general urn model of Hill, Lane and Sudderth. Interestingly for $k>2$ we found a critical dependence from the initial conditions beyond a certain values of the memory parameter $p$, and regions of convergence with entropy that is sub-linear in the number of steps.

math.PR↗

arXiv2025

Large deviations for Generalized Polya Urns with non-binary increments

In this paper we show how to extend the Sample-Path Large Deviation Principle for the urn model of Hill, Lane and Sudderth to the case in which the increment of the urn is not a binary variable. In particular, we sketch how to modify the Theorem 1 given in [Stochastic Processes and their Applications 127 (2017) 3372-3411] to include also urn processes with increments taking more than two values.

math.PR↗

arXiv2025

The Urn of Hill, Lane and Sudderth

We review some facts, properties and applications of the urn of Hill, Lane and Sudderth, a paradigmatic model of stochastic process with memory where the urn evolution is as follows: consider an urn of given capacity, at each step a new ball, black or white, is added to the urn with probability that is function (urn function) of the fraction of black balls. The process runs until capacity is reached.

math.PR↗

arXiv2025

Blanket representation for spin networks with n-body interactions

This paper extends the Blanket representation of [Universal scaling limits for spin networks via martingale methods, Franchini, S., Proc. R. Soc. A, 481 (2025)] from systems with two-body interactions to multi-spin (or n-body) interactions. This generalization allows for the exploration of broader physical phenomena where higher order interactions are significant.

cond-mat.stat-mech↗

arXiv2025

On the thermodynamic limit of bipartite spin networks

We investigate the properties of the thermodynamic limit in a general bipartite spin network with pairwise interactions. This is done by integrating one of the the spin groups, to transform the bipartite problem into a single group problem with a non-linear Hamiltonian. The transformed model is also relevant due to similarity with the LCVAE architecture.

cond-mat.stat-mech↗

arXiv2025

Causal horizon from quantum fluctuations

We propose a simple model of quantum void where the flow of time is deduced directly from quantum fluctuations and the consequent particle-antiparticle creations. Given a certain number of space-like separated pair creation events, assumed to happen all at the same initial time, we show that past and future can be foliated into a sequence of adapted manifolds based on how many events causally influence them. We also give an explicit construction for the simplest case of one space dimension.

cond-mat.stat-mech↗

arXiv2024

Catene ideali con numero fissato di auto-intersezioni

In this thesis we study in detail the self-intersection properties of Random Walks. Although notoriously hard to tackle, these properties are crucially related to the excluded-volume effect and other central features of real polymers. Our main purpose will be to study ideal chains (Random Walks) on lattice where the ratio between the number of self-intersections and the total length is fixed to some number.

cond-mat.stat-mech↗

arXiv2024

P-adic numbers and kernels

We discuss the relation between p-adic numbers and kernels in view of a recent large deviation theory for mean-field spin glasses. As an application we show several fundamental properties of numerical bases in kernel language. In particular, we show that the Derrida's Generalized Random Energy Model can be interpreted as a (random) numerical base. We also show an application to the Primon gas and the Riemann Zeta Function by constructing a kernel representation of the Primon gas based on a finite p-base, thereby establishing a concrete link between number theory and kernel theory.

cond-mat.dis-nn↗

arXiv2024

Universal scaling limits for spin networks via martingale methods

We use simple martingale methods to construct a large deviation theory of spin systems with pairwise interactions. As an application, we show that the fully connected case obeys a universal scaling limit that is just a product of magnetisation eigenstates.

cond-mat.stat-mech↗

arXiv2024

Lattice physics approaches for neural networks

Modern neuroscience has evolved into a frontier field that draws on numerous disciplines, resulting in the flourishing of novel conceptual frames primarily inspired by physics and complex systems science. Contributing in this direction, we recently introduced a mathematical framework to describe the spatiotemporal interactions of systems of neurons using lattice field theory, the reference paradigm for theoretical particle physics. In this note, we provide a concise summary of the basics of the theory, aiming to be intuitive to the interdisciplinary neuroscience community. We contextualize our methods, illustrating how to readily connect the parameters of our formulation to experimental variables using well-known renormalization procedures. This synopsis yields the key concepts needed to describe neural networks using lattice physics. Such classes of methods are attention-worthy in an era of blistering improvements in numerical computations, as they can facilitate relating the observation of neural activity to generative models underpinned by physical principles.

q-bio.NC↗

arXiv2023

A simplified Parisi Ansatz II: REM universality

In a previous work [A simplified Parisi Ansatz, Franchini, S., Commun. Theor. Phys., 73, 055601 (2021)] we introduced a simple method to compute the Random Overlap Structure of Aizenmann, Simm and Stars and the full RSB Parisi formula for the Sherrington-Kirkpatrick Model without using replica theory. The method consists in partitioning the system into smaller sub-systems that we call layers, and iterate the Bayes rule. A central ansatz in our derivation was that these layers could be approximated by Random Energy Models of the Derrida type. In this paper we analyze the properties of the interface in detail, and show the equivalence with the Random Energy Model at any temperature.

cond-mat.stat-mech↗

arXiv2023

Neural activity in quarks language: Lattice Field Theory for a network of real neurons

Brain-computer interfaces surged extraordinary developments in recent years, and a significant discrepancy now exists between the abundance of available data and the limited headway made in achieving a unified theoretical framework. This discrepancy becomes particularly pronounced when examining the collective neural activity at the micro- and meso-scale, where a coherent formalization that adequately describes neural interactions is still lacking. Here, we introduce a mathematical framework to analyze systems of natural neurons and interpret the related empirical observations in terms of lattice field theory, an established paradigm from theoretical particle physics and statistical mechanics. Our methods are tailored to interpret data from chronic neural interfaces, especially spike rasters from measurements of single neurons activity, and generalize the maximum entropy model for neural networks so that also the time evolution of the system is taken into account. This is obtained by bridging particle physics and neuroscience, paving the way to particle physics-inspired models of neocortex.

q-bio.NC↗

arXiv2022

Large Deviations Theory of Increasing Returns

An influential theory of increasing returns has been proposed by the economist W. B. Arthur in the '80s to explain the lock-in phenomenon between two competing commercial products. In the most simplified situation there are two competing products that gain customers according to a majority mechanism: each new customer arrives and asks which product they bought to a certain odd number of previous customers, and then buy the most shared product within this sample. It is known that one of these two companies reaches monopoly almost surely in the limit of infinite customers. Here we consider a generalization [G. Dosi, Y. Ermoliev, Y. Kaniovsky, J. Math. Econom. 23, 1-19 (1994)] where the new customer follows the indication of the sample with some probability, and buy the other product otherwise. Other than economy, this model can be reduced to the urn of Hill, Lane and Sudderth, and includes several models of physical interest as special cases, like the Elephant Random Walk, the Friedman's urn and other generalized urn models. We provide a large deviation analysis of this model at the sample-path level, and give a formula that allows to find the most likely trajectories followed by the market share variable. Interestingly, in the parameter range where the lock-in phase is expected, we observe a whole region of convergence where the entropy cost is sub-linear. We also find a non-linear differential equation for the cumulant generating function of the market share variable, that can be studied with a suitable perturbations theory.

math.PR↗