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Hugo Touchette

Publications and source records attributed to Hugo Touchette.

At least 19 recordsLinked to original sources
arXiv2025

Linear approximations of large deviations: Cubic diffusion test

We propose a method for approximating the large deviation rate function of time-integrated observables of diffusion processes, used in statistical physics to characterize the fluctuations of nonequilibrium systems. The method is based on linearizing the effective process associated with the large deviations of the process and observable considered, and is tested for a simple one-dimensional nonlinear diffusion model involving a cubic drift. The results show that the linear approximation compares well with the exact rate function, especially in the large fluctuation regime, and that its accuracy is related to the way the linearized process localizes in space. Possible extensions and applications to more complex diffusion models are proposed for future work.

cond-mat.stat-mech↗
arXiv2025

An introduction to large deviations with applications in physics

These notes are based on the lectures that one of us (HT) gave at the Summer School on the "Theory of Large Deviations and Applications", held in July 2024 at Les Houches in France. They present the basic definitions and mathematical results that form the theory of large deviations, as well as many simple motivating examples of applications in statistical physics, which serve as a basis for the many other lectures given at the school that covered more specific applications in biophysics, random matrix theory, nonequilibrium systems, geophysics, and the simulation of rare events, among other topics. These notes extend the lectures, which can be accessed online, by presenting exercises and pointer references for further reading.

cond-mat.stat-mech↗
arXiv2023

Large deviations of the stochastic area for linear diffusions

The area enclosed by the two-dimensional Brownian motion in the plane was studied by L\'evy, who found the characteristic function and probability density of this random variable. For other planar processes, in particular ergodic diffusions described by linear stochastic differential equations (SDEs), only the expected value of the stochastic area is known. Here, we calculate the generating function of the stochastic area for linear SDEs, which can be related to the integral of the angular momentum, and extract from the result the large deviation functions characterising the dominant part of its probability density in the long-time limit, as well as the effective SDE describing how large deviations arise in that limit. In addition, we obtain the asymptotic mean of the stochastic area, which is known to be related to the probability current, and the asymptotic variance, which is important for determining from observed trajectories whether or not a diffusion is reversible. Examples of reversible and irreversible linear SDEs are studied to illustrate our results.

cond-mat.stat-mech↗
arXiv2022

Dynamical large deviations of linear diffusions

Linear diffusions are used to model a large number of stochastic processes in physics, including small mechanical and electrical systems perturbed by thermal noise, as well as Brownian particles controlled by electrical and optical forces. Here, we use techniques from large deviation theory to study the statistics of time-integrated functionals of linear diffusions, considering three classes of functionals or observables relevant for nonequilibrium systems which involve linear or quadratic integrals of the state in time. For these, we derive exact results for the scaled cumulant generating function and the rate function, characterizing the fluctuations of observables in the long-time limit, and study in an exact way the set of paths or effective process that underlies these fluctuations. The results gives a complete description of how fluctuations arise in linear diffusions in terms of effective forces that remain linear in the state or, alternatively, in terms of fluctuating densities and currents that solve Riccati-type equations. We illustrate these results using two common nonequilibrium models, namely, transverse diffusions in two dimensions involving a non-conservative rotating force, and two interacting particles in contact with heat baths at different temperatures.

cond-mat.stat-mech↗
arXiv2022

Adaptive power method for estimating large deviations in Markov chains

We study the performance of a stochastic algorithm based on the power method that adaptively learns the large deviation functions characterizing the fluctuations of additive functionals of Markov processes, used in physics to model nonequilibrium systems. This algorithm was introduced in the context of risk-sensitive control of Markov chains and was recently adapted to diffusions evolving continuously in time. Here we provide an in-depth study of the convergence of this algorithm close to dynamical phase transitions, exploring the speed of convergence as a function of the learning rate and the effect of including transfer learning. We use as a test example the mean degree of a random walk on an Erd\"os-R\'enyi random graph, which shows a transition between high-degree trajectories of the random walk evolving in the bulk of the graph and low-degree trajectories evolving in dangling edges of the graph. The results show that the adaptive power method is efficient close to dynamical phase transitions, while having many advantages in terms of performance and complexity compared to other algorithms used to compute large deviation functions.

cond-mat.stat-mech↗
arXiv2022

Noise correction of large deviations with anomalous scaling

We present a path integral calculation of the probability distribution associated with the time-integrated moments of the Ornstein-Uhlenbeck process that includes the Gaussian prefactor in addition to the dominant path or instanton term obtained in the low-noise limit. The instanton term was obtained recently [D. Nickelsen, H. Touchette, Phys. Rev. Lett. 121, 090602 (2018)] and shows that the large deviations of the time-integrated moments are anomalous in the sense that the logarithm of their distribution scales nonlinearly with the integration time. The Gaussian prefactor gives a correction to the low-noise approximation and leads us to define an instanton variance giving some insights as to how anomalous large deviations are created in time. The results are compared with simulations based on importance sampling, extending our previous results based on direct Monte Carlo simulations. We conclude by explaining why many of the standard analytical and numerical methods of large deviation theory fail in the case of anomalous large deviations.

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arXiv2021

Learning nonequilibrium control forces to characterize dynamical phase transitions

Sampling the collective, dynamical fluctuations that lead to nonequilibrium pattern formation requires probing rare regions of trajectory space. Recent approaches to this problem based on importance sampling, cloning, and spectral approximations, have yielded significant insight into nonequilibrium systems, but tend to scale poorly with the size of the system, especially near dynamical phase transitions. Here we propose a machine learning algorithm that samples rare trajectories and estimates the associated large deviation functions using a many-body control force by leveraging the flexible function representation provided by deep neural networks, importance sampling in trajectory space, and stochastic optimal control theory. We show that this approach scales to hundreds of interacting particles and remains robust at dynamical phase transitions.

cond-mat.stat-mech↗
arXiv2021

Role of current fluctuations in nonreversible samplers

It is known that the distribution of nonreversible Markov processes breaking the detailed balance condition converges faster to the stationary distribution compared to reversible processes having the same stationary distribution. This is used in practice to accelerate Markov chain Monte Carlo algorithms that sample the Gibbs distribution by adding nonreversible transitions or non-gradient drift terms. The breaking of detailed balance also accelerates the convergence of empirical estimators to their ergodic expectation in the long-time limit. Here, we give a physical interpretation of this second form of acceleration in terms of currents associated with the fluctuations of empirical estimators using the level 2.5 of large deviations, which characterises the likelihood of density and current fluctuations in Markov processes. Focusing on diffusion processes, we show that there is accelerated convergence because estimator fluctuations arise in general with current fluctuations, leading to an added large deviation cost compared to the reversible case, which shows no current. We study the current fluctuation most likely to arise in conjunction with a given estimator fluctuation and provide bounds on the acceleration, based on approximations of this current. We illustrate these results for the Ornstein-Uhlenbeck process in two dimensions and the Brownian motion on the circle.

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arXiv2021

Large deviations of currents in diffusions with reflective boundaries

We study the large deviations of current-type observables defined for Markov diffusion processes evolving in smooth bounded regions of $\mathbb{R}^d$ with reflections at the boundaries. We derive for these the correct boundary conditions that must be imposed on the spectral problem associated with the scaled cumulant generating function, which gives, by Legendre transform, the rate function characterizing the likelihood of current fluctuations. Two methods for obtaining the boundary conditions are presented, based on the diffusive limit of random walks and on the Feynman--Kac equation underlying the evolution of generating functions. Our results generalize recent works on density-type observables, and are illustrated for an $N$-particle single-file diffusion on a ring, which can be mapped to a reflected $N$-dimensional diffusion.

cond-mat.stat-mech↗
arXiv2020

Dynamical large deviations of reflected diffusions

We study the large deviations of time-integrated observables of Markov diffusions that have perfectly reflecting boundaries. We discuss how the standard spectral approach to dynamical large deviations must be modified to account for such boundaries by imposing zero-current conditions, leading to Neumann or Robin boundary conditions, and how these conditions affect the driven process, which describes how large deviations arise in the long-time limit. The results are illustrated with the drifted Brownian motion and the Ornstein-Uhlenbeck process reflected at the origin. Other types of boundaries and applications are discussed.

cond-mat.stat-mech↗
arXiv2020

Efficient large deviation estimation based on importance sampling

We present a complete framework for determining the asymptotic (or logarithmic) efficiency of estimators of large deviation probabilities and rate functions based on importance sampling. The framework relies on the idea that importance sampling in that context is fully characterized by the joint large deviations of two random variables: the observable defining the large deviation probability of interest and the likelihood factor (or Radon-Nikodym derivative) connecting the original process and the modified process used in importance sampling. We recover with this framework known results about the asymptotic efficiency of the exponential tilting and obtain new necessary and sufficient conditions for a general change of process to be asymptotically efficient. This allows us to construct new examples of efficient estimators for sample means of random variables that do not have the exponential tilting form. Other examples involving Markov chains and diffusions are presented to illustrate our results.

cond-mat.stat-mech↗
arXiv2018

A Generalized Multifractal Formalism for the Estimation of Nonconcave Multifractal Spectra

Multifractal analysis has become a powerful signal processing tool that characterizes signals or images via the fluctuations of their pointwise regularity, quantified theoretically by the so-called multifractal spectrum. The practical estimation of the multifractal spectrum fundamentally relies on exploiting the scale dependence of statistical properties of appropriate multiscale quantities, such as wavelet leaders, that can be robustly computed from discrete data. Despite successes of multifractal analysis in various real-world applications, current estimation procedures remain essentially limited to providing concave upper-bound estimates, while there is a priori no reason for the multifractal spectrum to be a concave function. This work addresses this severe practical limitation and proposes a novel formalism for multifractal analysis that enables nonconcave multifractal spectra to be estimated in a stable way. The key contributions reside in the development and theoretical study of a generalized multifractal formalism to assess the multiscale statistics of wavelet leaders, and in devising a practical algorithm that permits this formalism to be applied to real-world data, allowing for the estimation of nonconcave multifractal spectra. Numerical experiments are conducted on several synthetic multifractal processes as well as on a real-world remote-sensing image and demonstrate the benefits of the proposed multifractal formalism over the state of the art.

math.FA↗
arXiv2018

Dynamical phase transition in drifted Brownian motion

We study the occupation fluctuations of drifted Brownian motion in a closed interval, and show that they undergo a dynamical phase transition in the long-time limit without an additional low-noise limit. This phase transition is similar to wetting and depinning transitions, and arises here as a switching between paths of the random motion leading to different occupations. For low occupations, the motion essentially stays in the interval for some fraction of time before escaping, while for high occupations the motion is confined in an ergodic way in the interval. This is confirmed by studying a confined version of the model, which points to a further link between the dynamical phase transition and quantum phase transitions. Other variations of the model, including the geometric Brownian motion used in finance, are considered to discuss the role of recurrent and transient motion in dynamical phase transitions.

cond-mat.stat-mech↗
arXiv2018

Large deviations of random walks on random graphs

We study using large deviation theory the fluctuations of time-integrated functionals or observables of the unbiased random walk evolving on Erd\"os-R\'enyi random graphs, and construct a modified, biased random walk that explains how these fluctuations arise in the long-time limit. Two observables are considered: the sum of the degrees visited by the random walk and the sum of their logarithm, related to the trajectory entropy. The modified random walk is used for both quantities to explain how sudden changes in degree fluctuations, akin to dynamical phase transitions, are related to localization transitions. For the second quantity, we also establish links between the large deviations of the trajectory entropy and the maximum entropy random walk.

cond-mat.stat-mech↗
arXiv2018

Spectral properties of simple classical and quantum reset processes

We study the spectral properties of classical and quantum Markovian processes that are reset at random times to a specific configuration or state with a reset rate that is independent of the current state of the system. We demonstrate that this simple reset dynamics causes a uniform shift in the eigenvalues of the Markov generator, excluding the zero mode corresponding to the stationary state, which has the effect of accelerating or even inducing relaxation to a stationary state. Based on this result, we provide expressions for the stationary state and probability current of the reset process in terms of weighted sums over dynamical modes of the reset-free process. We also discuss the effect of resets on processes that display metastability. We illustrate our results with two classical stochastic processes, the totally asymmetric random walk and the one-dimensional Brownian motion, as well as two quantum models: a particle coherently hopping on a chain and the dissipative transverse field Ising model, known to exhibit metastability.

cond-mat.stat-mech↗
arXiv2018

Adaptive sampling of large deviations

We introduce and test an algorithm that adaptively estimates large deviation functions characterizing the fluctuations of additive functionals of Markov processes in the long-time limit. These functions play an important role for predicting the probability and pathways of rare events in stochastic processes, as well as for understanding the physics of nonequilibrium systems driven in steady states by external forces and reservoirs. The algorithm uses methods from risk-sensitive and feedback control to estimate from a single trajectory a new process, called the driven process, known to be efficient for importance sampling. Its advantages compared to other simulation techniques, such as splitting or cloning, are discussed and illustrated with simple equilibrium and nonequilibrium diffusion models.

cond-mat.stat-mech↗
arXiv2018

Anomalous scaling of dynamical large deviations

The typical values and fluctuations of time-integrated observables of nonequilibrium processes driven in steady states are known to be characterized by large deviation functions, generalizing the entropy and free energy to nonequilibrium systems. The definition of these functions involves a scaling limit, similar to the thermodynamic limit, in which the integration time $\tau$ appears linearly, unless the process considered has long-range correlations, in which case $\tau$ is generally replaced by $\tau^\xi$ with $\xi\neq 1$. Here we show that such an anomalous power-law scaling in time of large deviations can also arise without long-range correlations in Markovian processes as simple as the Langevin equation. We describe the mechanism underlying this scaling using path integrals and discuss its physical consequences for more general processes.

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