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# Workshop YEP XVII: "Interacting Particle Systems"

## Aug 30 - Sep 3

#### Format

The workshop will take place in the week Aug 30 - Sep 3, 2021.

The event will be online.

#### Summary

The theory of Interacting Particle Systems focuses on the dynamics of systems consisting of a large or infinite number of entities, in which the mechanism of evolution is random and follows simple, local rules.
The topic had its beginnings in the 70s and 80s, motivated by Statistical Physics and fundamental problems from Probability Theory. It has since developed into a fruitful source of interesting mathematical questions and a very successful framework to model emerging collective complex behavior for systems in a variety of fields, including Biology, Economics and Social Sciences.
The field is sufficiently diverse that there are sub- communities that approach it from very different perspectives. We have identified three broad lines of interest –duality, scaling limits and invariant measures– and have used these to guide selection of speakers. Our goal is to provide an occasion for exchange of experiences and perspectives for young researchers in an immersive week.

#### Organizers

 Conrado da Costa Durham University Richard Kraaij TU Delft Federico Sau IST Austria, Klosterneuburg Daniel Valesin Groningen University

#### Speakers

Mini-courses

 Patrícia Gonçalves IST Lisbon Jan Swart Czech Academy of Science Cristina Toninelli Paris Dauphine

Overview Lectures

 Pablo Ferrari UBA Buenos Aires Frank Redig TU Delft

Invited Talks

 Márton Balázs Bristol University Oriane Blondel Université Lyon 1 Peter Mörters Bath University Tom Mountford EPFL Ellen Saada Université de Paris Alexandre Stauffer Università Roma Tre

Short Talks

 Luisa Andreis Università degli Studi di Firenze Leandro Chiarini IMPA/Utrecht University Simone Floreani TU Delft Chiara Franceschini IST Lisbon Bart van Ginkel TU Delft Jessica Jay University of Bristol Shubhamoy Nandan Leiden University Assaf Shapira Roma Tre University Réka Szabó Université Paris Dauphine Siamak Taati American University of Beirut Mario Ayala Valenzuela INRAE - Avignon Clare Wallace University of Durham

#### Abstracts

Luisa Andreis

Large deviations for sparse inhomogeneous random graphs and coagulation processes
Inhomogeneous random graphs are a natural generalization of the well-known Erdös–Rényi random graph, where vertices are characterized by a type and edges are independent but distributed according to the type of the vertices that they are connecting. In the sparse regime, these graphs undergo a phase transition in terms of the emergence of a giant component exactly as the clas-sical Erdös–Rényi model. In this talk we will present an alternative approach,via large deviations, to prove this phase transition. This allows a comparison with the gelation phase transition that characterizes some coagulation process and with phase transitions of condensation type emerging in several systems of interacting components.
(ongoing joint work with Wolfgang König (WIAS and TU Berlin),Tejas Iyer (WIAS), Heide Langhammer (WIAS), Robert Patterson (WIAS))

Márton Balázs

Queues, stationarity, and stabilisation of last passage percolation

Take a point x on the 2-dimensional integer lattice and another one y North-East from the first. Place i.i.d. Exponential weights on the vertices of the lattice; the last passage time between the two points is the maximal sum of these weights which can be collected by a path that takes North and East steps. The process of these weights as y varies is a difficult one, but locally has a nice asymptotic structure. I'll explain what stationary queues have to do with this and how this insight gives coalescence properties of the maximal-weight paths of last passage.
(joint work with Ofer Busani and Timo Seppäläinen)

PRESENTATION

Oriane Blondel

Kinetically constrained models out of equilibrium
Kinetically constrained models are interacting particle systems on Z^d, in which particles can appear/disappear only if a given local constraint is satisfied. This condition complexifies significantly the dynamics. In particular, it deprives the system of monotonicity properties, which leaves us with few tools to study the dynamics when it is initially not at equilibrium. I will review the results and techniques we have in this direction.

Leandro Chiarini

Stochastic homogenisation of Gaussian fields
In this talk, we discuss the convergence of a sequence of random fields that generalise the Gaussian Free Field and bi-Laplacian field. Such fields are defined in terms of non-homogeneous elliptic operators which will be sampled at random. Under standard assumptions of stochastic homogenisation, we identify the limit fields as the usual GFF and bi-Laplacian fields up to a multiplicative constant.

Pablo Ferrari

Gaussian random permutation and the free Bose gas
PDF

Simone Floreani

Self-duality in the continuum
We generalize the notion of stochastic self-duality beyond the framework of lattice systems. Our results are formulated as two self-intertwining relations for consistent particle systems: one related to falling factorial polynomials (classical self-dualities), the other one to generalized Charlier/Meixner polynomials and multiple-stochastic integrals (orthogonal self-dualities). Relevant examples include the Brownian gas, systems of sticky Brownian motions, free Kawasaki dynamics and a generalized inclusion process in the continuum.
(joint (ongoing) work with S. Jansen (LMU Munich), F. Redig (TU Delft) and S. Wagner (LMU Munich))

Chiara Franceschini

Duality for interacting particle systems via scalar product
In this talk I will briefly overview the concept of duality for Markov jump processes and I will present a theorem which shows how duality functions can be found relying on the scalar product with respect to the reversible measure of the process. This method works for symmetric, asymmetric and (work in progress) multi-species particle system. The theorem presented is written here for symmetric particles [1] and here in the context of asymmetric particles [2].

[1] G. Carinci, C. F., C. Giardinà, W. Groenevelt, F. Redig. “orthogonal dualities of Markov processes and unitary symmetries” SIGMA (2019) 15, 053
[2] G. Carinci, C. F., W. Groenevelt “q-orthogonal dualities for asymmetric particle systems” EJP (2021) 26, 1-38

Bart van Ginkel

The Symmetric Exclusion Process on a manifold
In this short talk I will introduce the Symmetric Exclusion Process on a compact Riemannian manifold and present our results on its Hydrodynamic Limit and Equilibrium Fluctuations. Moreover, I will highlight the challenges that come up when defining and studying this particle system in curved space.
(joint work with Frank Redig)

Patrícia Gonçalves

Scaling limits for symmetric exclusion with open boundary
In this mini-course I will describe the derivation of certain laws that rule the space-time evolution of the density of exclusion processes when put in contact with reservoirs. The main goal is to describe the connection between the macroscopic (continuous) equations and the microscopic (discrete) system of particles. The former can be either PDEs or stochastic PDEs depending on whether one is looking at the law of large numbers or the central limit theorem scaling.
I will focus on symmetric rates and I will explain how, depending on the finiteness of the variance of the transition probability, to obtain a collection of reaction-diffusion equations given in terms of the usual Laplacian  or the regional fractional Laplacian and with different types of boundary conditions. I will discuss the role and impact of the reservoirs' dynamics at the level of boundary conditions that appear at the macro level, ie, on the PDE.  I will also explain what is known for the fluctuations around the hydrodynamical profile  and what are the difficulties that one faces when dealing with systems with an open boundary.

Jessica Jay

Interacting Particle Systems and Jacobi Style Identities
In 2018, Balázs and Bowen gave a probabilistic proof of the Jacobi triple product identity, a well-known classical identity appearing throughout Mathematics and Physics. The proof follows by the exclusion-zero range correspondence through the stationary blocking measures of the asymmetric simple exclusion and zero range processes.
Naturally, one asks if other systems give rise to identities with combinatorial significance, via their stationary blocking measures. In this talk we consider another family of nearest neighbour interacting particle systems on the integer line and show that it gives rise to new combinatorial identities.
(joint work with Márton Bálazs and Dan Fretwell)

Peter Mörters

Percolation phase transition in weight-dependent random connection models
We investigate spatial random graphs defined on the points of a Poisson process in d-dimensional space, which combine scale-free degree distributions and long-range effects. Every Poisson point is assigned an independent weight. Given the weight and position of the points, we form an edge between any pair of points independently with a probability depending on the two weights of the points and their distance. Preference is given to short edges and connections to vertices with large weights. We characterize the parameter regime where there is a nontrivial percolation phase transition and show that it depends not only on the power-law exponent of the degree distribution but also on a geometric model parameter. Based on joint work with Peter Gracar and Lukas Lüchtrath.

Tom Mountford

The Invariance principle for Recurrent Markov Cookie Random walks
We show that in the recurrent case, when diffusively rescaled, the Markov cookie process converges in law to a Brownian motion perturbed at extremities.  The key idea is to consider a coarse graining of the process and to appeal to ray Knight theorems.
(joint work with E. Kosygina and J. Peterson)

Shubhamoy Nandan

Spatially Inhomogeneous Populations with Seed-banks
We construct an interacting particle system (IPS) that describes the genetic evolution over time of populations with seed-banks. The colonies are spatially located on the integer lattice Z^d, d\ge 1, and each colony consists of two populations: active and dormant(seed-bank). The sizes of the active and the dormant populations (seed-banks) are finite, vary across different colonies and remain fixed throughout the evolution of the IPS over time. Individuals carry one of the two genetic types: A and B, and change type via resampling as long as they are active. Active individuals in each colony can also exchange type with individuals in the constituent seed-bank. Active individuals resample not only from their own (active) population, but also from active population of other colonies according to a random walk transition kernel. The latter is referred to as migration. We show that under a mild condition on the sizes of the active populations,the IPSis well-defined and has a dual. The dual consists of a system of interacting coalescing random walks in an inhomogeneous environment that switch between an active state and a dormant state.The IPS converges to a unique equilibrium that depends on the initial density of types, and exhibits a dichotomy between clustering(monotype equilibrium) and coexistence(multi-type equilibrium). This dichotomy is determined by a clustering criterion: clustering occurs if and only if two random walks in the dual starting from arbitrary states eventually coalesce with probability one.Further,we show that if the relative strengths of the seed-banks in different colonies are uniformly bound, then the latter is equivalent to the symmetrized migration kernel being recurrent. Joint work with Frank den Hollander.
References:

Frank Redig

1 - Duality for interacting particle systems: introduction to the algebraic approach,
2- Applications of duality in macroscopic fields

1 - I will introduce the concept of duality for Markov processes and its relation to symmetries of the associated generator. I will explain how from natural Lie algebraic objects one can construct Markov generators with symmetries and associated dualities, and provide several examples in conservative particle systems such as the symmetric exclusion and inclusion process. We will treat both “classical dualities” (factorial moments) and orthogonal polynomial dualities.

2 - I will provide several applications of duality (and especially orthogonal polynomial duality) in the study of macroscopic fields of interacting particle systems. First we will see how the hydrodynamic limit is related to the scaling limit of one particle. Second we will see how the Boltzmann Gibbs principle can be understood and quantified via orthogonal polynomial duality, and finally we will see how higher order fluctuation fields (generalizing the density fluctuation field) can be constructed via orthogonal polynomial duality.
The material is mostly based on the manuscript in preparation “Duality for Markov processes, a Lie algebraic approach” jointly with G. Carinci, C Giardina.
(macroscopic field material is based on joint works with M. Ayala, G. Carinci, F. Sau, S. Floreani)

PRESENTATION

Invariant measures for multilane exclusion process
We consider the simple exclusion process on $\Z\times\{0,1\}$, that is,  an “horizontal  ladder” composed of two lanes. Particles can jump according to a lane-dependent translation-invariant nearest neighbour jump kernel, i.e. “horizontally” along each lane, and “vertically” along the scales of the ladder. We analyze the set of extremal invariant measures for this model.
(joint work with Gidi Amir, Christophe Bahadoran and Ofer Busani. ArXiv : 2105.12974)

Assaf Shapira

Diffusive scaling of the Kob-Andersen model

The Kob-Andersen model is an interacting particle system on the lattice, in which sites can contain at most one particle. Each particle is allowed to jump to an empty neighboring site only if there are sufficiently many empty sites in its neighborhood. This way, when the density is very high, the time it takes for the system to evolve is very long. We will see that the system still scales diffusively, but with a coefficient which decays extremely fast as the density increases.

Alexandre Stauffer

Non-equilibrium multi-scale analysis and coexistence in competing first-passage percolation
We consider a natural random growth process with competition on Z^d called first-passage percolation in a hostile environment, that consists of two first-passage percolation processes  FPP_1 and FPP_\lambda that compete for the occupancy of sites. Initially FPP_1 occupies the  origin and spreads through the edges of Z^d at rate 1, while FPP_\lambda is initialised at sites  called seeds that are distributed according to a product of Bernoulli measures of parameter p. A seed remains dormant until FPP_1 or FPP_\lambda attempts to occupy it, after which  it spreads through the edges of Z^d at rate \lambda. We will discuss the results known for  this model and present a recent proof that the two types can coexist (concurrently produce an infinite  cluster) on Z^d. We remark that, though counterintuitive, the above model is not monotone in the sense that adding a seed of  FPP_\lambda could favor FPP_1. A central contribution of our work is the development of a novel  multi-scale analysis to analyze this model, which we call a multi-scale analysis with non-equilibrium feedback and which we believe could help analyze other models with non-equilibrium dynamics and lack of monotonicity.
(joint work with Tom Finn (Univ. of Bath))

Jan Swart

Interacting Particle Systems: Almost sure uniqueness, pathwise duality, and the mean-field limit
In this mini-course, I will discuss the almost surely unique construction of interacting particle systems from a graphical representation, pathwise duality, and the mean-field limit. The study of pathwise duals in the mean-field limit naturally leads to recursive tree processes and the question of endogeny. Recursive tree processes also occur naturally in the study of frozen percolation, which is an interacting particle system with long-range dependence to which the standard existence and uniqueness results do not apply and for which the question of almost sure uniqueness turns out to be rather subtle.
The course is based on Chapters 4 and 6 of my lecture notes on interacting particle systems (available at http://staff.utia.cas.cz/swart/YEP.html).
(joint work with Niklas Latz, Tibor Mach, Balázs Ráth, Anja Sturm, Márton Szőke, and Tamás Terpai)

Réka Szabó

From survival to extinction of the contact process by the removal of a single edge
The contact process is usually seen as a model of epidemics:  vertices are individuals, which can be healthy or infected; infected individuals recover with rate 1 and transmit the infection to each neighbour with rate lambda. One can study how local changes on the underlying graph affect relevant quantities associated to the model. In this talk we give a construction of a tree in which the contact process with any positive infection rate survives but, if a certain privileged edge is removed, one obtains two subtrees in which the contact process with infection rate smaller than 1/4 dies out. (joint work with Daniel Valesin)

Siamak Taati

Positive-rate interacting particle systems with Bernoulli invariant measures
Every positive-rate finite-range interacting particle system (in discrete/continuous time) that has a Bernoulli invariant measure is ergodic.  The proof is via the entropy method, but unlike the usual entropy arguments, does not require shift-invariance of the starting measure.  Two new (?) ingredients are a sharp bound on the entropy increase and a "bootstrap" argument.  As an application, this result imposes a sever limitation on the computation power of reversible cellular automata in presence of noise.
(joint work Irène Marcovici)

Cristina Toninelli

Kinetically Constrained Spin Models

Kinetically Constrained Spin Models (KCSM) are interacting particle systems on Z^d which have been introduced by physicists in the 80’s to model the liquid/glass transition and more generally the "glassy behaviour" occurring in a large variety of systems (colloidal suspensions, vibrated granulars, emulsions, foams...). Each lattice site is either empty or occupied by a particle and the evolution is given by a continuous time Markov process with elementary moves corresponding to the creation/destruction of particles. The key feature is that a move can occur only if the configuration verifies a local constraint which specifies the maximal number of particles in a certain neighbourhood.

Extensive numerical simulations indicate that, for proper choices of the constraints, KCSM display a remarkable glassy behavior which makes them particularly appealing for physicists striving for the long-lived open problem of the liquid/glass transition. At the same time, from a mathematical point of view, KCSM pose very challenging and interesting issues. In fact, the hardness of the constraints induce non-attractiveness, the occurrence of several invariant measures, and the failure of many powerful tools (coercive inequalities, coupling, censoring...). Most importantly, the degeneracy of the rates is not a mere technical obstacle which prevents the use of classic tools. Indeed, the behavior of KCSM is qualitatively different from that of other Glauber dynamics. Peculiar features include: anomalously long mixing times, aging, singularities in the dynamical large deviation function, dynamical heterogeneities, and ergodicity breaking transitions leading to a large variety of amorphous structures.

The aim of these lectures is to review the existing mathematical results on KCSM. In particular, we will illustrate recent results on the large time behavior of the stationary process, including the full universality picture in two dimensions. On the way, we will explain the connection of KCSM with another well known class of models : bootstrap percolation cellular automata. Finally, we will present a choice of open problems concerning the out of equilibrium dynamics of KCSM. Indeed, despite some achievements, robust tools are still lacking to analyse the models in this regime and several beautiful questions remain open.

PRESENTATION

Mario Ayala Valenzuela

A spatial stochastic model for a vector-borne virus population: some scaling limits
We build a measure-valued stochastic process representing the dynamics of a virus population, structured by phenotypic traits and geographical space, and where viruses are transported between spatial locations by mechanical vectors. By combining various scaling limits on population sizes, speed of diffusion of vectors, and other relevant model parameters, we show the emergence of a system of integro-differential equations as scaling limit.
(work in progress with Jérôme Coville (BioSP, INRAE)  and Raphaël Forien (BioSP, INRAE))

Clare Wallace

A Conditional Renewal Random Walk and the Ising model
We consider the trajectories of a renewal random walk, that is, a random walk on the two-dimensional integer lattice whose jumps have positive horizontal component.
We prove conditional versions of the Functional Law of Large Numbers and the Functional Central Limit Theorem for these trajectories, when they are placed under large-deviations conditions on the terminal height and integral of the trajectories. We find the shape of the limiting profile of the trajectories, and discuss the convergence of the distribution of the fluctuations around this profile to that of a conditioned Gaussian process.
Finally, we discuss the connection between this random walk and macroscopic sections of the phase boundary in the two-dimensional, low-temperature Ising model.

#### Poster session

Alberto Montefusco

Variational structures behind second-order hyperbolic PDEs

The subject of this poster is based on the following two ideas:

1. In 1974, Kac proposed a stochastic model where the probability density solves the telegrapher’s equation in one dimension [1], a second-order hyperbolic PDE that is formally identical to the hyperbolic heat equation proposed by Cattaneo [2].
2. In 2014, Mielke, Peletier and Renger showed that the variational structure associated with a given dissipative dynamics may be inferred from the large deviations of an underlying stochastic model [3].

In our work, we calculate the large deviations of Kac’s model and find a variational structure behind the telegrapher’s equation. We compare this variational structure with the typical one that is proposed for second-order hyperbolic systems, for instance in [4, Sec. 5.4].

Then, following [5, 6], we aim to exploit the variational formulation to prove that the telegrapher’s equation converges, in the overdamped limit, to a parabolic PDE.

Several open questions stem from this work. How can the picture be extended to more than one dimension? Can we find an interacting particle system behind heat conduction? Second-order damped hyperbolic models like the telegrapher’s equation usually show undesirable properties (e.g., the non-negativity of a probability density is not preserved) in more than one dimension, especially in the short-time regime. Can such models be used at intermediate time scales, before the corresponding parabolic model becomes applicable? How would the underlying stochastic particle systems look like?
References:

[1] M. Kac. “A stochastic model related to the telegrapher’s equation”. In: Rocky Mountain Journal of Mathematics 4.3 (1974), pp. 497–510.

[2] C. Cattaneo. “Sulla Conduzione Del Calore”. In: Atti del Seminario Matematico e Fisico dell’Universitá di Modena e Reggio Emilia 3.3 (1948).

[3] A. Mielke, M. A. Peletier, and D. R. M. Renger. “On the Relation between Gradient Flows and the Large-Deviation Principle, with Applications to Markov Chains and Diffusion”. In: Potential Analysis 41.4 (2014), pp. 1293–1327.

[4] M. Pavelka, V. Klika, and M. Grmela. Multiscale Thermo-Dynamics. Introduction to GENERIC . Berlin/Boston: De Gruyter, 2018.

[5] M. H. Duong, A. Lamacz, M. A. Peletier, and U. Sharma. “Variational approach to coarse-graining of generalized gradient flows”. In: Calculus of Variations and Partial Differential Equations 56.4 (2017).

[6] B. Hilder, M. A. Peletier, U. Sharma, and O. Tse. “An inequality connecting entropy distance, Fisher Information and large deviations”. In: Stochastic Processes and their Applications 130.5 (2020), pp. 2596– 2638.

Michele Aleandri

Delay-induced periodic behaviour in competitive populations

We study a model of binary decisions in a fully connected network of interacting agents. Individual decisions are determined by social influence, coming from direct interactions with neighbours, and a group level pressure that accounts for social environment. We study the convergence of the process as the number of agents goes to infinity and the propagation of chaos. Moreover, when the number of agents is large but fixed, we show the amount of time spent by the process around the stable points of the macroscopic dynamics. As a consequence, in a competitive environment, the interplay of these two aspects results in the presence of a persistent disordered phase where no majority is formed.
We show how the introduction of a delay mechanism in the agent's detection of the global average choice may drastically change this scenario, giving rise to a coordinated self sustained periodic behaviour.  When the delay mechanism is led by kernels of Erlang type the limit dynamics is described by a finite number of equations. In the competitive environment tuning the parameters of the delay an Hops bifurcation occurs and limit cycles appear. Through the paper we show the evolution of the microscopic process performing some simulations.

Andrej Srakar

Approximating the Ising model on fractal lattices of dimension two or greater than two
The exact solutions of the Ising model in one and two dimensions are well known, but much less is known about solutions on fractal lattices. In an important contribution, Codello, Drach and Hietanen (2015) construct periodic approximations to the free energies of Ising models on fractal lattices of dimension smaller than two using a generalization of combinatorial method of Feynman (1972) and Vdovichenko (1965). We generalize their approach to fractal lattices of dimensions 2 and greater than 2, in particular of Koch curve variety (e.g. quadratic and von Koch surface). To this end we combine combinatorial optimization and transfer matrix approaches, referring to earlier works of Andrade and Salinas (1984). We compute approximate estimates for the critical temperatures and compare them to more usual Monte Carlo estimates. Similar as in Codello et al., we compute the correlation length as a function of the temperature and extract the relative critical exponent. The method allows generalizations to any fractal lattice, as well as concrete solutions to approach solutions for other non-translationally invariant lattices (e.g. those with random interactions).

Lucas R. de Lima

Limiting shape for the Frog Model on abelian groups of polynomial growth

We study conditions for the existence of the asymptotic shape for random processes defined on Cayley graphs of finitely generated groups with polynomial growth. We will focus our attention on the cases of the Frog Model on abelian groups and on an ongoing study for a class of subadditive processes. We set the Frog model to start with a particle at each vertex of the graph, and only one of these particles is active when the process begins. Each activated particle performs a simple random walk in discrete time activating the inactive particles in the visited vertices.
The considered class of graphs is an algebraic generalization of the hypercubic Z^d lattice, and the related limiting shape results combine probability with techniques from geometric group theory (joint work with Cristian Coletti).

Rowel Gündlach

Invasion Percolation on Power Law GW Trees
We consider the invasion percolation cluster (IPC) on a weighted power law Galton-Watson tree. Such a cluster can be obtained by taking the subtree T_N that is obtained after performing N steps of Prim’s algorithm, starting from the root (ø). The invasion percolation cluster is then defined by T(ø) = lim_{N\to \infty} T_N . The IPC is a one-ended tree that consists of a unique path to innity (backbone) onto which finite clusters are attached. We also denote the k-cut IPC, which is the cluster containing the root when the edge between the k-th and (k + 1)-th backbone vertex is removed.
The poster highlights three results. First, we construct the k-cut IPC up to arbitrary depth, and by letting k \to \infty, are able to construct the IPC. Secondly, we consider the behaviour of the weights along the backbone. More precisely, the evolution of the future maximum added weight that under the appropriate scaling converges to a non-trivial stochastic process. Lastly, we present a volume scaling that describes the number of vertices found in the k-cut IPC, for k large.
The presented work fits into the literature as previous authors have touched upon the IPC on the regular tree [1] and the finite-variance Galton Watson tree [2], which are both captured in our model. The results from the poster are from a paper that is still in progress but currently has many interesting results and open questions that support an interesting discussion on this topic.

[1] Angel, O. et al. (2008). Invasion percolation on regular trees. Ann. Probab, 36(2):420–466

[2] Michelen, M., et al. (2019). Invasion percolation on Galton-Watson trees. Electron. J. Probab, 24(31):1–35

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Start:
Aug 30
End:
Sep 3
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