arXiv subjects

Federico Sau

Publications and source records attributed to Federico Sau.

27 records · Page 2Linked to original sources

arXiv2021

Orthogonal polynomial duality of boundary driven particle systems and non-equilibrium correlations

We consider symmetric partial exclusion and inclusion processes in a general graph in contact with reservoirs, where we allow both for edge disorder and well-chosen site disorder. We extend the classical dualities to this context and then we derive new orthogonal polynomial dualities. From the classical dualities, we derive the uniqueness of the non-equilibrium steady state and obtain correlation inequalities. Starting from the orthogonal polynomial dualities, we show universal properties of n-point correlation functions in the non-equilibrium steady state for systems with at most two different reservoir parameters, such as a chain with reservoirs at left and right ends.

math.PR↗

arXiv2021

Hydrodynamics for the partial exclusion process in random environment

In this paper, we introduce a random environment for the exclusion process in $\mathbb{Z}^d$ obtained by assigning a maximal occupancy to each site. This maximal occupancy is allowed to randomly vary among sites, and partial exclusion occurs. Under the assumption of ergodicity under translation and uniform ellipticity of the environment, we derive a quenched hydrodynamic limit in path space by strengthening the mild solution approach initiated in [39] and [21]. To this purpose, we prove, employing the technology developed for the random conductance model, a homogenization result in the form of an arbitrary starting point quenched invariance principle for a single particle in the same environment, which is a result of independent interest. The self-duality property of the partial exclusion process allows us to transfer this homogenization result to the particle system and, then, apply the tightness criterion in [40].

math.PR↗

arXiv2021

Symmetric inclusion process with slow boundary: hydrodynamics and hydrostatics

We study the hydrodynamic and hydrostatic limits of the one-dimensional open symmetric inclusion process with slow boundary. Depending on the value of the parameter tuning the interaction rate of the bulk of the system with the boundary, we obtain a linear heat equation with either Dirichlet, Robin or Neumann boundary conditions as hydrodynamic equation. In our approach, we combine duality and first-second class particle techniques to reduce the scaling limit of the inclusion process to the limiting behavior of a single, non-interacting, particle.

math.PR↗

arXiv2021

Higher order hydrodynamics and equilibrium fluctuations of interacting particle systems

Motivated by the recent preprint [arXiv:2004.08412] by Ayala, Carinci, and Redig, we first provide a general framework for the study of scaling limits of higher order fields. Then, by considering the same class of infinite interacting particle systems as in [arXiv:2004.08412], namely symmetric simple exclusion and inclusion processes in the d-dimensional Euclidean lattice, we prove the hydrodynamic limit, and convergence for the equilibrium fluctuations, of higher order fields. In particular, the limit fields exhibit a tensor structure. Our fluctuation result differs from that in [arXiv:2004.08412], since we consider a different notion of higher order fluctuation fields.

math.PR↗

arXiv2020

Symmetric simple exclusion process in dynamic environment: hydrodynamics

We consider the symmetric simple exclusion process in $\mathbb Z^d$ with quenched bounded dynamic random conductances and prove its hydrodynamic limit in path space. The main tool is the connection, due to the self-duality of the process, between the invariance principle for single particles starting from all points and the macroscopic behavior of the density field. While the hydrodynamic limit at fixed macroscopic times is obtained via a generalization to the time-inhomogeneous context of the strategy introduced in \cite{nagy_symmetric_2002}, in order to prove tightness for the sequence of empirical density fields we develop a new criterion based on the notion of uniform conditional stochastic continuity, following \cite{varadhan_stochasticprocesses_2007}. In conclusion, we show that uniform elliptic dynamic conductances provide an example of environments in which the so-called arbitrary starting point invariance principle may be derived from the invariance principle of a single particle starting from the origin. Therefore, our hydrodynamics result applies to the examples of quenched environments considered in, e.g.,\ \cite{andres2014}, \cite{andres_quenched_2018}, \cite{biskup_invariance_2019} in combination with the hypothesis of uniform ellipticity.

math.PR↗

arXiv2019

Stochastic duality and eigenfunctions

We start from the observation that, anytime two Markov generators share an eigenvalue, the function constructed from the product of the two eigenfunctions associated to this common eigenvalue is a duality function. We push further this observation and provide a full characterization of duality relations in terms of spectral decompositions of the generators for finite state space Markov processes. Moreover, we study and revisit some well-known instances of duality, such as Siegmund duality, and extract spectral information from it. Next, we use the same formalism to construct all duality functions for some solvable examples, i.e., processes for which the eigenfunctions of the generator are explicitly known.

math.PR↗

arXiv2018

Factorized duality, stationary product measures and generating functions

We find all factorized duality functions for a class of interacting particle systems. The functions we recover are self-duality functions for interacting particle systems such as zero-range processes, symmetric inclusion and exclusion processes, as well as duality and self-duality functions for their continuous counterparts. The approach is based on, firstly, a general relation between factorized duality functions and stationary product measures and, secondly, an intertwining relation provided by generating functions. For the interacting particle systems, these self-duality and duality functions turn out to be generalizations of those previously obtained in [9] and, more recently, in [8]. Thus, we discover that only these two families of dualities cover all possible cases. Moreover, the same method discloses all self-duality functions for interacting diffusion systems such as the Brownian energy process, where both the process and its dual are in continuous variables.

math.PR↗

arXiv2016

Generalized immediate exchange models and their symmetries

We reconsider the immediate exchange model and define a more general class of models where mass is split, exchanged and merged. We relate the splitting process to the symmetric inclusion process via thermalization and from that obtain symmetries and self-duality of the generalized IEM model. We show that analogous properties hold for models were the splitting is related to the symmetric exclusion process or to independent random walkers.

math.PR↗

arXiv2015

Duality and stationary distributions of the "Immediate Exchange Model" and its generalizations

We prove that the "Immediate Exchange Model" of econophysics has a discrete dual, where the duality functions are those connecting the Brownian Energy Process and the Symmetric Inclusion Process. As a consequence, we recover invariance of products of Gamma distributions with shape parameter 2, and obtain ergodicity results. Next we show similar properties of a generalized model, where the exchange fraction is $Beta(s,t)$ distributed (instead of uniform), and product measures with $\mbox{Gamma}(s+t)$ marginals are invariant. We prove that the discrete dual has the self-duality property, and prove full SU(1,1) for both the continuous and discrete model.

math.PR↗