[Paper Review] A basic introduction to large deviations: Theory, applications, simulations

Hugo Touchette|arXiv (Cornell University)|Jun 21, 2011

Advanced Thermodynamics and Statistical Mechanics45 references47 citations

TL;DR

This paper provides a foundational introduction to large deviation theory, emphasizing intuitive understanding and practical applications in statistical physics, stochastic processes, and simulation. It presents key concepts like the rate function, scaled cumulant generating function, and the Gärtner-Ellis theorem, while introducing numerical methods such as importance sampling and exponential change of measure for estimating rare event probabilities in systems like Markov chains and stochastic differential equations.

ABSTRACT

The theory of large deviations deals with the probabilities of rare events (or fluctuations) that are exponentially small as a function of some parameter, e.g., the number of random components of a system, the time over which a stochastic system is observed, the amplitude of the noise perturbing a dynamical system or the temperature of a chemical reaction. The theory has applications in many different scientific fields, ranging from queuing theory to statistics and from finance to engineering. It is also increasingly used in statistical physics for studying both equilibrium and nonequilibrium systems. In this context, deep analogies can be made between familiar concepts of statistical physics, such as the entropy and the free energy, and concepts of large deviation theory having more technical names, such as the rate function and the scaled cumulant generating function. The first part of these notes introduces the basic elements of large deviation theory at a level appropriate for advanced undergraduate and graduate students in physics, engineering, chemistry, and mathematics. The focus there is on the simple but powerful ideas behind large deviation theory, stated in non-technical terms, and on the application of these ideas in simple stochastic processes, such as sums of independent and identically distributed random variables and Markov processes. Some physical applications of these processes are covered in exercises contained at the end of each section. In the second part, the problem of numerically evaluating large deviation probabilities is treated at a basic level. The fundamental idea of importance sampling is introduced there together with its sister idea, the exponential change of measure. Other numerical methods based on sample means and generating functions, with applications to Markov processes, are also covered.

Motivation & Objective

• To provide advanced undergraduate and graduate students in physics, engineering, and mathematics with an intuitive, non-technical introduction to large deviation theory.
• To bridge concepts in statistical physics—such as entropy and free energy—with their counterparts in large deviation theory, including the rate function and scaled cumulant generating function.
• To introduce numerical methods for estimating large deviation probabilities, focusing on importance sampling and exponential change of measure.
• To offer practical exercises linking theory to real-world applications in stochastic processes, including Markov chains and stochastic differential equations.
• To motivate further study by highlighting the theoretical depth and computational utility of large deviation theory across diverse scientific fields.

Proposed method

• Uses simple, non-technical language to explain core ideas of large deviation theory, focusing on sums of i.i.d. random variables and Markov processes.
• Applies the Gärtner-Ellis theorem to derive the rate function from the scaled cumulant generating function (SCGF), assuming existence of probability densities.
• Introduces importance sampling and exponential change of measure as central techniques for efficient simulation of rare events.
• Employs the Metropolis-Hastings algorithm to sample from tilted probability distributions, enabling numerical evaluation of large deviation probabilities.
• Proposes the sample mean method and empirical generating function method for estimating rate functions from simulated trajectories.
• Connects path large deviations to the largest eigenvalue of a tilted generator, enabling numerical computation via eigenvalue solvers.

Experimental results

Research questions

• RQ1 How can the probability of rare fluctuations in stochastic systems be quantified using large deviation theory?
• RQ2 What is the relationship between statistical mechanics concepts like entropy and free energy and large deviation counterparts such as the rate function and SCGF?
• RQ3 How can numerical methods like importance sampling and exponential change of measure be used to efficiently estimate large deviation probabilities?
• RQ4 What are the optimal paths in stochastic processes that lead to rare fluctuations, and how can they be computed?
• RQ5 How do numerical methods such as the sample mean and empirical SCGF methods converge for bounded and unbounded random variables?

Key findings

• The rate function for sums of i.i.d. random variables can be derived using the Gärtner-Ellis theorem, with the SCGF serving as the Legendre-Fenchel transform of the rate function.
• For Markov chains and continuous-time processes, the path large deviation principle is linked to the largest eigenvalue of a tilted generator, enabling numerical computation.
• Importance sampling with exponential change of measure significantly improves the efficiency of rare event simulations by biasing the sampling toward rare fluctuations.
• The empirical SCGF method converges quickly for bounded random variables like the Bernoulli case, while convergence slows for unbounded variables such as the exponential distribution.
• Optimal paths for rare fluctuations in conservative systems are time-reversed versions of the noiseless decay dynamics, providing a geometric interpretation of large deviations.
• For additive processes under Langevin dynamics, the optimal path to a fluctuation $ S_T = s $ is the constant path $ x(t) = s $, consistent with the asymptotic solution of the boosted SDE.