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An introduction to large deviations with applications in physics

Ivan N. Burenev, Daniël W. H. Cloete, Vansh Kharbanda, Hugo Touchette

TL;DR

The notes establish large deviation theory as a framework for quantifying rare fluctuations in complex systems, introducing the large deviation principle and the scaled cumulant generating function λ(k) as central objects. They develop the Gärtner–Ellis theorem and Legendre–Fenchel duality to obtain rate functions I(a) from λ(k) without requiring full distributional knowledge, and illustrate these ideas through IID sums, Sanov's theorem, and Markovian dynamics (discrete and continuous time) as well as Markov diffusions. The text further explains how tilted generators and the Feynman–Kac formalism yield SCGFs for time-integrated observables, and how symmetrisation and the generalized Doob transform define effective processes that realise conditioned large deviations in the long-time limit. Together with exercises and further readings, the work provides a practical, methodical toolkit for analyzing nonequilibrium fluctuations, rare transitions, and the spectral structures underlying large deviations in physical systems.

Abstract

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.

An introduction to large deviations with applications in physics

TL;DR

The notes establish large deviation theory as a framework for quantifying rare fluctuations in complex systems, introducing the large deviation principle and the scaled cumulant generating function λ(k) as central objects. They develop the Gärtner–Ellis theorem and Legendre–Fenchel duality to obtain rate functions I(a) from λ(k) without requiring full distributional knowledge, and illustrate these ideas through IID sums, Sanov's theorem, and Markovian dynamics (discrete and continuous time) as well as Markov diffusions. The text further explains how tilted generators and the Feynman–Kac formalism yield SCGFs for time-integrated observables, and how symmetrisation and the generalized Doob transform define effective processes that realise conditioned large deviations in the long-time limit. Together with exercises and further readings, the work provides a practical, methodical toolkit for analyzing nonequilibrium fluctuations, rare transitions, and the spectral structures underlying large deviations in physical systems.

Abstract

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.

Paper Structure

This paper contains 36 sections, 2 theorems, 167 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Basics of large deviation theory
  3. Applications of large deviations
  4. Simple examples of IID sample means
  5. Large deviation principle
  6. Varadhan's lemma
  7. Gärtner--Ellis theorem
  8. Legendre--Fenchel transform
  9. Properties of λ(k) and I(a)
  10. Exercises
  11. Further reading
  12. Simple applications
  13. Cramér's theorem
  14. Sanov's theorem
  15. Exercises
  16. ...and 21 more sections

Key Result

Theorem 1

Let $(A_n)_{n>0}$ be a sequence of RVs defined on some (nice) space $\mathcal{A}$ and satisfying the LDP with a (nice) rate function $I(a)$. For any real and bounded function $f:\mathcal{A}\rightarrow\mathbb{R}$ of $A_n$, we have The $\sup$ in this expression is a generalised maximum that can return $\infty$.

Figures (12)

  • Figure 1: Sample mean of Gaussian random variables with $\mu=1$ and $\sigma=1$. Left: Probability density $P(S_n=s)$. Right: Convergence of the LDP limit to the rate function $I(s)$ (dashed curve).
  • Figure 2: Properties of the LF transform $f^*$ of a function $f$ and the double LF transform $f^{**}$.
  • Figure 3: Duality between the slopes and values of a function $f$ and its LF transform $f^*$.
  • Figure 4: Left: Probability density of an exponentially-distributed RV. Right: Rate function of the corresponding sample mean.
  • Figure 5: SCGF (left) and rate function (right) for the sample mean of $n$ Bernoulli RVs, plotted for $p=1/2$.
  • ...and 7 more figures

Theorems & Definitions (19)

  • Example 1
  • Example 2
  • Definition 1: Intuitive LDP
  • Definition 2: More mathematical LDP
  • Definition 3: Complete LDP
  • Theorem 1: Varadhan 1966
  • Theorem 2: Gärtner 1977, Ellis 1984
  • Example 3
  • Example 4
  • Example 5
  • ...and 9 more