Hugo Touchette


My research covers many areas of applied mathematics and theoretical physics, driven by an unhealthy interest in anything random and unpredictable: noisy dynamical systems, turbulence, coin tossing, finance, the weather, etc.

I'm trained as a physicist but now work mostly at the interface of probability theory, statistics, simulation, optimization, control, and, more recently, machine learning.

My main specialty is the theory of large deviations - a branch of probability theory used to estimate the probability of very rare events arising in random systems as diverse as gases, queues, random walks, information systems or nonequilibrium systems driven by noise and external forces.

I have written a review article on the many applications of large deviation theory in statistical physics. I also have lecture notes and a webpage on the subject.

The following are projects I'm currently working on:
In previous years, I also worked on Brownian motion with solid friction, studied the physics of information in feedback control, and got distracted by football graphs.

Possible student projects

Suitable for all levels: Honors, MSc, PhD and post-doc. Interested students are encouraged to contact me for more information or to discuss other projects.

Fluctuation dynamics and process conditioning

The study of how fluctuations arise in a stochastic process - e.g., how a queue overflows or how a financial market crashes - is fundamental for predicting and controlling their behavior. In physics, fluctuations are also known to be connected to the response of systems to perturbations and external forces.

The idea of this project is to see how a Markov process 'builds up' fluctuations in time by conditioning (in a probabilistic sense) that process on observing a given fluctuation and by describing this conditioning as a new Markov process, called the effective or fluctuation process.

Large deviations for random walks on graphs

This an application of the previous project focused on using conditioning on rare events to find network characteristics of random graphcs, such as connected components, typical regions, central nodes, etc. A related project is to study escape-type problems of random walks evolving on random graphs in the limit of large graphs.

Numerical methods for large deviations

Large deviations are difficult to probe numerically as they are, by definition, rarely seen in direct simulations (think of a meteorite hitting earth). The aim of this project is to develop efficient numerical methods, based on Monte-Carlo simulations, to 'accelerate' the sampling of large deviations or rare events in general in Markov processes. This is an active area in physics, statistics, engineering, and financial mathematics.

Dynamical phase transitions

Dynamical phase transitions are sudden changes that arise in the fluctuations of stochastic processes or, more precisely, in the mechanisms that create fluctuations. The goal of this project is to study dynamical phase transitions in Markov processes and, especially, in stochastic differential equations. Analogies with phas transitions arising in physics, e.g., in solids and liquids at a certain critical temperature, are established via large deviation theory.

Stochastic processes with reset

Markov processes jumping at random from one state to another (depending on their current state) are used in physics to model equilibrium and nonequilibrium systems perturbed by noise. Recently, there has been a lot of interest on a new class of Markov processes, called reset processes, evolving stochastically for some time and then jumping to a fixed 'reset' point. The goal of this project is to develop the theory of these processes and, here again, their large deviation properies.


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