Numerical Aspects of Large Deviations

TL;DR

Numerical Aspects of Large Deviations introduces a general framework to compute tail probabilities in stochastic processes via large-deviation sampling. The core idea is to separate the randomness from the rest of the model by representing it with a vector of uniform random numbers and to sample biased ensembles using an exponential tilt with a temperature-like parameter $\Theta$, followed by unbiasing to recover the true distribution $Q(S)$. The authors illustrate the approach on the Bernoulli process and demonstrate how to obtain tails down to probabilities as small as $10^{-160}$, then extend the method to arbitrary stochastic processes and a broad set of applications, including random walks, non-interacting fermions, traffic, disease spread, and stochastic thermodynamics. They also discuss practical aspects such as normalization constants $Z(\Theta)$, histogram stitching across multiple $\Theta$, computational resources, and alternative rare-event strategies, arguing that this framework provides a unifying toolkit for large-deviation studies in both equilibrium and non-equilibrium settings.

Abstract

An introduction to numerical large-deviation sampling is provided. First, direct biasing with a known distribution is explained. As simple example, the Bernoulli experiment is used throughout the text. Next, Markov chain Monte Carlo (MCMC) simulations are introduced. In particular, the Metropolis-Hastings algorithm is explained. As first implementation of MCMC, sampling of the plain Bernoulli model is shown. Next, an exponential bias is used for the same model, which allows one to obtain the tails of the distribution of a measurable quantity. This approach is generalized to MCMC simulations, where the states are vectors of $U(0,1)$ random entries. This allows one to use the exponential or any other bias to access the large-deviation properties of rather arbitrary random processes. Finally, some recent research applications to study more complex models are discussed.

Figures (9)

• Figure 1: Numerically measured histograms of the number $l$ of 1's obtained from $n=50$ coin flips. The histograms are measured from $M=10^4$, $10^6$ and $10^8$ experiments. The solid line shows the analytical solution Eq. (\ref{['eq:binomial']}).
• Figure 2: Histograms of the number of 1's obtained from $n=50$ coin flips for four different values $\beta$ of the probability to obtain a 1, rescaled to result in the distribution for $\alpha=0.3$. The histograms are obtained from $M=10^4$ numerical experiments, respectively. The solid line shows the analytic solution Eq. (\ref{['eq:binomial']}).
• Figure 3: Flow of probability from and to state $y$. This and other states are shown as circles. It is assumed that $y$ has non-zero transition probabilities to and from three other states $z_1$, $z_2$ and $z_3$, shown as arrows. Other possible transitions of the states $z_1$, $z_2$ and $z_3$ are not shown here.
• Figure 4: Example for a non ergodic system. Possible transitions with non-zero transition probabilities are indicated by arrows. A Markov chain which starts in $y$=W or $y=$ Y will always have $P$(X,$t$)$=0$ and $P$(Z,$t$)$=0$ while this will not be the case if the chain started in X or Z. Thus, the limiting distribution will depend on the initial state.
• Figure 5: Sample Markov chain for the Bernoulli case for $n=50$ coin flips: Number $l(t)=\sum_i y_i(t)$ of 1's as function of the Monte Carlo step $t$ for probability $\alpha=0.5$ and $n_{\rm c}=2$. Two extreme different initial configurations are taken into account. The horizontal line indicates the expectation value $\alpha n$.