arXiv subjects

Cristian Giardinà

Publications and source records attributed to Cristian Giardinà.

At least 19 recordsLinked to original sources

arXiv2026

The boundary-driven multispecies harmonic process

We introduce the multispecies version of the harmonic process on a one-dimensional chain in contact with boundary reservoirs. This process is a continuous-time Markov chain where each site can host an unbounded number of colored particles. The symmetric bulk dynamics is put in contact with reservoirs, which inject and remove particles driving the system out-of-equilibrium. The Markov generator of the process is identified with the integrable Hamiltonian of an open rational Heisenberg spin chain of higher rank. We derive the underlying R- and K-matrices in operator form and construct the double-row transfer matrix following Sklyanin. Similar to the monospecies case the R-matrix factorises into two R-operators, each factor corresponding to left and right moving particles. We further define three dual models: an absorbing particle model, a hidden parameter model and a heat conduction model.

math-ph↗

arXiv2026

On a $Λ$-mutation model and the harmonic model

We introduce a continuous-time mutation model with two types determined by a finite measure $Λ$ on the unit interval. The model satisfies a certain consistency property known from mathematical population genetics and includes so called harmonic models being of interest in mathematical statistical physics. We mainly focus on the situation when the number of particles is equal to some constant $N$. Duality results and scaling limits as $N\to\infty$ for the forward and backward processes are provided leading to a commutative diagram. The stationary distribution of the forward process is studied with an emphasis on the case when $Λ$ is a beta distribution. The work bridges particle models from mathematical statistical physics and mutation models from mathematical population genetics.

math.PR↗

arXiv2026

Uphill transport in competitive drift-diffusion models with volume exclusion

This paper addresses uphill transport (defined as a regime in which particle flow is opposite to the prescriptions of Fick's diffusion) in drift-diffusion particle transport constrained by volume exclusion. Firstly, we show that the stationary hydrodynamic limit of a multispecies, weakly asymmetric exclusion process (SHDL) naturally predicts precisely characterized uphill regimes in the space of external drivings. Then, with specific reference to systems of oppositely charged particles, we identify well-defined model hypotheses and extensions whereby the SHDL converges to the modified Poisson-Nernst-Planck model, thus bridging the gap between exclusion-based particle models and continuum descriptions commonly used in engineering. The merits and limitations of the models in describing the particle fluxes and predicting uphill transport conditions are investigated in detail with respect to the adopted approximations and simplifications. The results demonstrate the persistence of uphill transport phenomena across modeling scales, clarify the conditions under which they occur, and suggest that uphill transport may play a significant role in nanoscale electrolytes, confined ionic and iontronic devices, and membrane-based technologies.

cond-mat.stat-mech↗

arXiv2025

Annealed Potts models on rank-1 inhomogeneous random graphs

In this paper, we study the annealed ferromagnetic $q$-state Potts model on sparse rank-1 random graphs, where vertices are equipped with a vertex weight, and the probability of an edge is proportional to the product of the vertex weights. In an annealed system, we take the average on both numerator and denominator of the ratio defining the Boltzmann-Gibbs measure of the Potts model. We show that the thermodynamic limit of the pressure per particle exists for rather general vertex weights. In the infinite-variance weight case, we show that the critical temperature equals infinity. For finite-variance weights, we show that, under a rather general condition, the phase transition is {\em first order} for all $q\geq 3$. However, we cannot generally show that the discontinuity of the order parameter is {\em unique}. We prove this uniqueness under a reasonable condition that holds for various distributions, including uniform, gamma, log-normal, Rayleigh and Pareto distributions. Further, we show that the first-order phase transition {\em persists} even for some small positive external field. In the rather relevant case of Pareto distributions with power-law exponent $τ$, remarkably, the phase transition is first order when $τ\geq 4$, but not necessarily when the weights have an infinite third-moment, i.e., when $τ\in(3,4)$. More precisely, the phase transition is second order for $τ\in (3,τ(q)]$, while it is first order when $τ>τ(q)$, where we give an explicit equation that $τ(q)$ solves.

math.PR↗

arXiv2024

Intertwining and propagation of mixtures for generalized KMP models and harmonic models

We study a class of stochastic models of mass transport on discrete vertex set $V$. For these models, a one-parameter family of homogeneous product measures $\otimes_{i\in V} ν_θ$ is reversible. We prove that the set of mixtures of inhomogeneous product measures with equilibrium marginals, i.e., the set of measures of the form \[ \int\Big(\bigotimes_{i\in V} ν_{θ_i}\Big) \,Ξ(\prod_{i\in V}dθ_i) \] is left invariant by the dynamics in the course of time, and the ``mixing measure'' $Ξ$ evolves according to a Markov process which we then call ``the hidden parameter model''. This generalizes results from [7] to a larger class of models and on more general graphs. The class of models includes discrete and continuous generalized KMP models, as well as discrete and continuous harmonic models. The results imply that in all these models, the non-equilibrium steady state of their reservoir driven version is a mixture of product measures where the mixing measure is in turn the stationary state of the corresponding ``hidden parameter model''. For the boundary-driven harmonic models on the chain $\{1,\ldots, N\}$ with nearest neighbor edges, we recover that the stationary measure of the hidden parameter model is the joint distribution of the ordered Dirichlet distribution (cf. [3]), with a purely probabilistic proof based on a spatial Markov property of the hidden parameter model.

math.PR↗

arXiv2024

Non-compact quantum spin chains as integrable stochastic particle processes

In this paper we discuss a family of models of particle and energy diffusion on a one-dimensional lattice, related to those studied previously in [Sasamoto-Wadati], [Barraquand-Corwin] and [Povolotsky] in the context of KPZ universality class. We show that they may be mapped onto an integrable $\mathfrak{sl}(2)$ Heisenberg spin chain whose Hamiltonian density in the bulk has been already studied in the AdS/CFT and the integrable system literature. Using the quantum inverse scattering method, we study various new aspects, in particular we identify boundary terms, modeling reservoirs in non-equilibrium statistical mechanics models, for which the spin chain (and thus also the stochastic process) continues to be integrable. We also show how the construction of a "dual model" of probability theory is possible and useful. The fluctuating hydrodynamics of our stochastic model corresponds to the semiclassical evolution of a string that derives from correlation functions of local gauge invariant operators of $\mathcal{N}=4$ super Yang-Mills theory (SYM), in imaginary-time. As any stochastic system, it has a supersymmetric completion that encodes for the thermal equilibrium theorems: we show that in this case it is equivalent to the $\mathfrak{sl}(2|1)$ superstring that has been derived directly from $\mathcal{N}=4$ SYM.

math-ph↗

arXiv2024

Exact solution of an integrable non-equilibrium particle system

We consider the integrable family of symmetric boundary-driven interacting particle systems that arise from the non-compact XXX Heisenberg model in one dimension with open boundaries. In contrast to the well-known symmetric exclusion process, the number of particles at each site is unbounded. We show that a finite chain of $N$ sites connected at its ends to two reservoirs can be solved exactly, i.e. the factorial moments of the non-equilibrium steady-state can be written in closed form for each $N$. The solution relies on probabilistic arguments and techniques inspired by integrable systems. It is obtained in two steps: i) the introduction of a dual absorbing process reducing the problem to a finite number of particles; ii) the solution of the dual dynamics exploiting a symmetry obtained from the Quantum Inverse Scattering Method. Long-range correlations are computed in the finite-volume system. The exact solution allows to prove by a direct computation that, in the thermodynamic limit, the system approaches local equilibrium. A by-product of the solution is the algebraic construction of a direct mapping between the non-equilibrium steady state and the equilibrium reversible measure.

math-ph↗

arXiv2024

Solvable stationary non equilibrium states

We consider the one dimensional boundary driven harmonic model and its continuous version, both introduced in \cite{FGK}. By combining duality and integrability the authors of \cite{FG} obtained the invariant measures in a combinatorial representation. Here we give an integral representation of the invariant measures which turns out to be a convex combination of inhomogeneous product of geometric distributions for the discrete model and a convex combination of inhomogeneous product of exponential distributions for the continuous one. The mean values of the geometric and of the exponential variables are distributed according to the order statistics of i.i.d. uniform random variables on a suitable interval fixed by the boundary sources. The result is obtained solving exactly the stationary condition written in terms of the joint generating function. The method has an interest in itself and can be generalized to study other models. We briefly discuss some applications.

math.PR↗

arXiv2023

Duality for the multispecies stirring process with open boundaries

We study the stirring process with $N-1$ species on a generic graph $G=(V,\mathcal{E})$ with reservoirs. The multispecies stirring process generalizes the symmetric exclusion process, which is recovered in the case $N=2$. We prove the existence of a dual process defined on an extended graph $\widetilde{G}=(\widetilde{V},\widetilde{\mathcal{E})}$ which includes additional extra-sites $\widetilde{V}\setminus V$ where dual particles get absorbed in the long-time limit. We thus obtain a characterization of the non-equilibrium steady state of the boundary-driven system in terms of the absorption probabilities of dual particles. The process is integrable for the case of the one-dimensional chain with two reservoirs at the boundaries and with maximally one particle per site. We compute the absorption probabilities by relying on the underlying ${gl}(N)$ symmetry and the matrix product ansatz. Thus one gets a closed-formula for (long-ranged) correlations and for the non-equilibrium stationary measure. Extensions beyond this integrable set-up are also discussed.

math-ph↗

arXiv2023

Large deviations and additivity principle for the open harmonic process

We consider the boundary driven harmonic model, i.e. the Markov process associated to the open integrable XXX chain with non-compact spins. Using the factorial moments we characterize the stationary measure as a mixture of product measures. For all spin values, we identify the law of the mixture in terms of the Dirichlet process. Next, by using the explicit knowledge of the non-equilibrium steady state we establish formulas predicted by Macroscopic Fluctuation Theory for several quantities of interest: the pressure (by Varadhan's lemma), the density large deviation function (by contraction principle), the additivity principle (by using the Markov property of the mixing law). To our knowledge, the results presented in this paper constitute the first rigorous derivation of these macroscopic properties for models of energy transport with unbounded state space, starting from the microscopic structure of the non-equilibrium steady state.

math.PR↗

arXiv2023

Density fluctuations for the multi-species stirring process

We study the density fluctuations at equilibrium of the multi-species stirring process, a natural multi-type generalization of the symmetric (partial) exclusion process. In the diffusive scaling limit, the resulting process is a system of infinite-dimensional Ornstein-Uhlenbeck processes that are coupled in the noise terms. This shows that at the level of equilibrium fluctuations the species start to interact, even though at the level of the hydrodynamic limit each species diffuses separately. We consider also a generalization to a multi-species stirring process with a linear reaction term arising from species mutation. The general techniques used in the proof are based on the Dynkin martingale approach, combined with duality for the computation of the covariances.

math.PR↗

arXiv2023

Integrable heat conduction model

We consider a stochastic process of heat conduction where energy is redistributed along a chain between nearest neighbor sites via an improper beta distribution. Similar to the well-known Kipnis-Marchioro-Presutti (KMP) model, the finite chain is coupled at its ends with two reservoirs that break the conservation of energy when working at different temperatures. At variance with KMP, the model considered here is integrable and one can write in a closed form the $n$-point correlation functions of the non-equilibrium steady state. As a consequence of the exact solution one can directly prove that the system is in a `local equilibrium' and described at the macro-scale by a product measure. Integrability manifests itself through the description of the model via the open Heisenberg chain with non-compact spins. The algebraic formulation of the model allows to interpret its duality relation with a purely absorbing particle system as a change of representation.

cond-mat.stat-mech↗

arXiv2021

Boundary driven Markov gas: duality and scaling limits

Inspired by the recent work of Bertini and Posta, who introduced the boundary driven Brownian gas on $[0,1]$, we study boundary driven systems of independent particles in a general setting, including particles jumping on finite graphs and diffusion processes on bounded domains in $\mathbb{R}^d$. We prove duality with a dual process that is absorbed at the boundaries, thereby creating a general framework that unifies dualities for boundary driven systems in the discrete and continuum setting. We use duality first to show that from any initial condition the systems evolve to the unique invariant measure, which is a Poisson point process with intensity the solution of a Dirichlet problem. Second, we show how the boundary driven Brownian gas arises as the diffusive scaling limit of a system of independent random walks coupled to reservoirs with properly rescaled intensity.

math.PR↗

arXiv2021

Annealed inhomogeneities in random ferromagnets

We consider spin models on complex networks frequently used to model social and technological systems. We study the annealed ferromagnetic Ising model for random networks with either independent edges (Erdős-Rényi), or with prescribed degree distributions (configuration model). Contrary to many physical models, the annealed setting is poorly understood and behaves quite differently than the quenched system. In annealed networks with a fluctuating number of edges, the Ising model changes the degree distribution, an aspect previously ignored. For random networks with Poissonian degrees, this gives rise to three distinct annealed critical temperatures depending on the precise model choice, only one of which reproduces the quenched one. In particular, two of these annealed critical temperatures are finite even when the quenched one is infinite, since then the annealed graph creates a giant component for all sufficiently small temperatures. We see that the critical exponents in the configuration model with deterministic degrees are the same as the quenched ones, which are the mean-field exponents if the degree distribution has finite fourth moment, and power-law-dependent critical exponents otherwise. Remarkably, the annealing for the configuration model with random i.i.d. degrees washes away the universality class with power-law critical exponents.

cond-mat.dis-nn↗

arXiv2021

Switching interacting particle systems: scaling limits, uphill diffusion and boundary layer

In this paper we consider three classes of interacting particle systems on $\mathbb Z$: independent random walks, the exclusion process, and the inclusion process. We allow particles to switch their jump rate (the rate identifies the type of particle) between $1$ (fast particles) and $ε\in[0,1]$ (slow particles). The switch between the two jump rates happens at rate $γ\in(0,\infty)$. In the exclusion process, the interaction is such that each site can be occupied by at most one particle of each type. In the inclusion process, the interaction takes places between particles of the same type at different sites and between particles of different type at the same site. We derive the macroscopic limit equations for the three systems, obtained after scaling space by $N^{-1}$, time by $N^2$, the switching rate by $N^{-2}$, and letting $N\to\infty$. The limit equations for the macroscopic densities associated to the fast and slow particles is the well-studied double diffusivity model. This system of reaction-diffusion equations was introduced to model polycrystal diffusion and dislocation pipe diffusion, with the goal to overcome the limitations imposed by Fick's law. In order to investigate the microscopic out-of-equilibrium properties, we analyse the system on $[N]=\{1,\ldots,N\}$, adding boundary reservoirs at sites $1$ and $N$ of fast and slow particles, respectively. Inside $[N]$ particles move as before, but now particles are injected and absorbed at sites $1$ and $N$ with prescribed rates that depend on the particle type. We compute the steady-state density profile and the steady-state current. It turns out that uphill diffusion is possible, i.e., the total flow can be in the direction of increasing total density. This phenomenon, which cannot occur in a single-type particle system, is a violation of Fick's law made possible by the switching between types.

math.PR↗

arXiv2021

Annealed Ising model on configuration models

In this paper, we study the annealed ferromagnetic Ising model on the configuration model. In an annealed system, we take the average on both sides of the ratio {defining the Boltzmann-Gibbs measure of the Ising model}. In the configuration model, the degrees are specified. Remarkably, when the degrees are deterministic, the critical value of the annealed Ising model is the same as that for the quenched Ising model. For independent and identically distributed (i.i.d.) degrees, instead, the annealed critical value is strictly smaller than that of the quenched Ising model. This identifies the degree structure of the underlying graph as the main driver for the critical value. Furthermore, in both contexts (deterministic or random degrees), we provide the variational expression for the annealed pressure. Interestingly, our rigorous results establish that only part of the heuristic conjectures in the physics literature were correct.

math.PR↗

arXiv2021

Approximating the cumulant generating function of triangles in the Erdös-Rényi random graph

We study the pressure of the "edge-triangle model", which is equivalent to the cumulant generating function of triangles in the Erdös-Rényi random graph. By analyzing finite graphs of increasing volume, as well as the graphon variational problem in the infinite volume limit, we locate a curve in the parameter space where a one-step replica symmetry breaking transition occurs. Sampling a large graph in the broken symmetry phase is well described by a graphon with a structure very close to the one of an equi-bipartite graph.

cond-mat.stat-mech↗

arXiv2020

Duality in quantum transport models

We develop the `duality approach', that has been extensively studied for classical models of transport, for quantum systems in contact with a thermal `Lindbladian' bath. The method provides (a) a mapping of the original model to a simpler one, containing only a few particles and (b) shows that any dynamic process of this kind with generic baths may be mapped onto one with equilibrium baths. We exemplify this through the study of a particular model: the quantum symmetric exclusion process introduced in [D. Bernard, T. Jin, Phys. Rev. Lett. 123, 080601 (2019)]. As in the classical case, the whole construction becomes intelligible by considering the dynamical symmetries of the problem.

cond-mat.stat-mech↗