MATH 7880 (Spring '13)
Large Deviations and Gibbs Measures

Time & Place: M 1:35-2:55 p.m. in LCB 222 and W 1:35-2:55 p.m. in JWB 333

Instructor: Firas Rassoul-Agha

Phone: (801) 585-1647, E-Mail: firas@math.utah.edu

Office Hour: M 9:50-10:50 a.m. or by email or by appointment, at LCB 209

The law of large numbers says that if we toss a fair coin a large number of times we will get about 50% heads. In fact, getting 50% heads is the most probable event. The central limit theorem, on the other hand, describes fluctuations from this event. It says that if we draw the histogram of a large number of tosses we will see the famous bell curve.

These basic limit theorems in probability theory allow us to understand the typical behavior of a stochastic system. However, a better understanding of the system can only be achieved when we also understand how atypical events can occur. This is particularly important when rare events have a high cost associated with them (e.g. earthquakes and other natural disasters).

Large deviation theory is an area of probability that seeks to quantify chances of extremely rare behavior, for example the kind that falls outside the central limit theorem. It has its origins in early probability and statistical mechanics. A unified formulation emerged in the 1960's especially through the work of S.R.S. Varadhan, who received the Abel Prize ("Nobel Prize of mathematics") for this and other achievements in 2007.

There are applications in various fields such as engineering, especially queueing theory, statistics and economics. Notions of entropy appear naturally in descriptions of these small probabilities.

Gibbs measures are probability measures that describe random systems in multidimensional space and possess a natural spatial Markovian property. They arose in the context of statistical physics.

This course intends to cover



Course lecture notes (January 10 version)

Supplementary Material:

Sister web page

Notes on Probability Theory (measure theory, weak convergence, LLN, CLT, LIL, conditioning, Markov chains, martingales, ergodic theorem, CLT for martingales, dynamic programming)

More notes on Probability Theory (Poisson processes, continuous time Markov processes [jump processes, semigroups, martingales, explossion, recurrence, transience], Brownian motion, diffusion processes, stochastic integration, Brownian motion on the half-line)

Notes on Stochastic Processes (Same as above notes on Brownian motion, diffusions, stochastic integration, and Brownian motion on the line. Also, stochastic differential equations, random time change, limits of Markov chains to stochastic processes, and reflected processes in higher dimensions)

Notes on Stochastic Processes (Same as above and has two lectures on Kolmogorov equations and invariant measures)

More notes on Stochastic Processes (different from the above. Stochastic integration, SDEs, exit problem, invariant distributions, LLN, CLT, convergence to equilibrium, large deviations, homogenization, scaling limits [and a lecture on log-Sobolev inequality])

Yet more notes on Stochastic Processes (Combination of topics from the probability and stochastic processes notes. Notes written more concisely)

Notes on Stochastic Calculus (Brownian motion, stochastic integration, and SDEs [as above]. Then, Gaussian space, Malliavin calculus, Hormander's theorem, stochastic partial differential equations, Navier-Stokes with noise)

Notes on Large Deviations (Cramer's theorem, Sanov's theorem, Schilder's theorem, LIL, small noise, exit problem, Markov chains, random walk among Bernoulli traps, hydrodynamic limits)

More notes on Large Deviations (more advanced, leading to hydrodynamic scaling of exclusion processes)

An invited paper on Large Deviations


(Also: notes on statistics)