Mario Ayala

Postdoctoral researcher · School of Computation, Information and Technology · TU München

How does macroscopic behavior emerge from microscopic randomness?

Portrait of Mario Ayala

Bienvenidos! My name is Mario Ayala.

My research is in probability theory, and more precisely in the rigorous passage from microscopic rules of interaction to macroscopic laws: how large collections of interacting particles generate collective behavior, how the fluctuations around this behavior can be characterized, and what survives when randomness is coarse-grained or reinterpreted. What I find most attractive about this subject is that the macroscopic law is never postulated: it is obtained, together with the precise sense in which the microscopic system converges to it.

Beyond formal research, I enjoy developing intuition through simulations, informal notes and the occasional interdisciplinary detour; you can play with the simulations yourself on the Simulations page.

The dots in the background are an actual symmetric exclusion process (SEP), the same class of models behind much of my research. Thanks a lot to Bart van Ginkel for his help getting them right.

Contact:

Research lines

My work develops along three lines that share one toolbox, i.e. self-duality and Dirichlet forms. The map shows how they connect; the three sections below describe each line.

I. Scaling limits of interacting particle systemsthe core
Duality and fluctuations
→ A duality theory at the fluctuation scale
Singular scaling limits
→ Fluctuations in the condensation regime → Spaces of varying dimension
Beyond the density field
→ A scaling theory of non-conserved observables
Self-duality
IRWCharlierSEPKrawtchoukSIPMeixner
𝔼η D(ξ, ηt) = 𝔼ξ D(ξt, η)
Dirichlet forms
Mosco convergence
↓ reversibility via Dirichlet forms
II. Effective dynamics
→ Fluctuating hydrodynamics without second moments
slow-fast elimination: right equilibria, wrong rare events
↓ duality, where the model allows it
III. Population models
→ Duality as a modelling instrument
JSP published submitted in prep. → open direction Titles link to arXiv, line names to the papers of each line.

I. Scaling limits of interacting particle systems

This is the core of my work, developed largely with Frank Redig (TU Delft) and Gioia Carinci (University of Modena and Reggio Emilia). My laboratory consists of three self-dual particle systems, i.e. independent random walkers, the symmetric exclusion process (SEP) and the symmetric inclusion process (SIP), whose duality functions are orthogonal polynomials (Charlier, Krawtchouk and Meixner respectively).

Duality reduces a many-particle question to a few-particle one, and with its help I study three things: the density fluctuation field together with its higher order analogues; singular limits, where Mosco convergence of Dirichlet forms is the right notion of convergence, such as condensation and, with Johannes Zimmer (TU München), the non-local hydrodynamics of SIP with long jumps; and the observables that the hydrodynamic limit discards, such as collisions and fluxes in SEP (with Michiel Renger, TU München). In work in preparation with Nicolas Dirr (Cardiff University) and Johannes Zimmer we also obtain the atomic limit of the Dean-Kawasaki equation.

II. Effective dynamics and coarse-graining

Most stochastic models are reduced descriptions: a Langevin equation is obtained by eliminating fast degrees of freedom, a hydrodynamic equation by averaging over particles, and a classical Markov chain by eliminating quantum coherences. The question behind this line is when such a reduced description is faithful to the dynamics it comes from.

With Nicolas Dirr, Grigorios Pavliotis (Imperial College London) and Johannes Zimmer we obtain conditions under which a diffusion with multiplicative noise is reversible, for every interpretation of the noise from Itô through Stratonovich to Klimontovich, and we show that reversibility is preserved under coarse-graining. With Benjamín Vallejo Jiménez (University of Colima) we ask the same question of the Merton consumption-investment problem, i.e. when the choice of stochastic integral changes the optimal policy and when it does not.

III. Stochastic population models

This is where my interest in stochastic processes started, as an undergraduate at the Mathematical and Theoretical Biology Institute (Arizona State University), working on chytridiomycosis in a harlequin frog population. Later, at INRAE, with Jérôme Coville, Raphaël Forien and Samuel Soubeyrand, I worked on measure-valued models that follow each individual of a population, for vector-borne viruses and for grouped dispersal.

With Johannes Zimmer I now study populations that cooperate through a shared medium, such as quorum sensing. The result I find most interesting so far is that, in a quorum-sensing reaction network, the speed of the signal leaves every mean-field equilibrium unchanged and nevertheless changes the barrier governing rare collapse, i.e. eliminating the signal first gives the right equilibria and the wrong rare events. In work in preparation we treat both the cells and the medium as interacting particle systems.

My Erdős number is 4: Me → Frank Redig → Pablo A. Ferrari → Peter E. Ney → Paul Erdős