18.177: FOUR PROBLEMS IN PROBABILITY AND MATHEMATICAL PHYSICS (TUESDAY/THURSDAY 2:30-4, 2-131)

This year's 18.177 course is a selective introduction to statistical physics, probability, and field theory. The course is structured around four unsolved problems, listed and explained below. These problems have received a fair amount of attention. Solutions to special cases of Problem 2 have led to two recent Fields medals, for example, and the Clay Institute has placed a million dollar bounty on Problem 4. I chose these particular problems because thinking about them gives us an excuse to internalize fundamental concepts that we would have wanted to learn anyway (ergodic theory, zero-one laws, Brownian motion, FKG inequalities, Ito calculus, the Gaussian free field and other random distributions, Schramm-Loewner evolution, etc.) and to study some exciting new results (Smirnov's recent work on percolation and the Ising model, for example). The existing literature on the problems is already quite rich, and the problems build on each other in a coherent way. For each of the four units, students taking the course for credit will be required to complete a problem set and provide a short summary of their own attempts to make progress (however limited) on the problem. It will be great if publishable research emerges from this course, but for an A grade it will suffice to give a good short exposition (couple of pages) of an existing result and an explanation of how one gets stuck when trying to apply it to the big problem. The progress summaries can be completed as group projects in collaboration with other students.

• 1. INFINITE CLUSTER IN CRITICAL 3D PERCOLATION

The first open problem is arguably the "very most basic question" about the "very simplest" statistical physics model of all.

Many statistical physics models involve random functions on the edges or vertices of a lattice with some (often quite sophisticated) probability distribution. The simplest is called percolation. It involves independently declaring each edge of a grid to be "open" with probability p and "closed" with probability 1-p. It is simple because everything is independent. One does not have to worry about complicated multi-point laws, which are very hard to work out in other contexts.

A fundamental question about percolation is whether, with positive probability, one can reach an infinite set of locations, starting from the origin, while staying on open edges. It is not too hard to show that there is a "critical value" p_c between 0 and 1 such the the answer is no if p is less than p_c and yes if p is greater than p_c. But what is the answer if p=p_c? This question has been answered for high dimensions (19 or above) and low dimensions (1 and 2). But despite half a century of research on percolation models, this question remains unanswered in dimension 3. The change that happens at p_c is called a "phase transition" and is the simplest case of a wide variety of interesting phase transitions in statistical physics.

• 2. SCHRAMM-LOEWNER EVOLUTION (SLE) AS A SCALING LIMIT OF RANDOM SELF-AVOIDING PATHS ON GRIDS

Brownian motion is a canonical model of a random path. One can draw a two dimensional Brownian motion by making each of the coordinates an independent one-dimensional Brownian motion. Such a path will cross itself many times with probability one.

What is the most canonical model of a random path in the plane that does NOT cross itself? The answer seems to be a particular family of random curves introduced by Oded Schramm in 1999 called Schramm-Loewner evolution (SLE) curves.

In a sense, all of the various random non-self-intersecting curves on grids that appear in so-called "critical" statistical physics models are believed to look approximately like SLE curves when the grid mesh is fine enough. We will focus on a particularly natural family of discrete random curves for which this has been formulated as a precise conjecture. The conjecture has been proved in several important special cases (including the case that the curves are boundaries of certain two-dimensional "critical percolation clusters"), but a comprehensive theory remains elusive.

• 3. LIOUVILLE QUANTUM GRAVITY (PLUS SLE) AS A SCALING LIMIT OF RANDOM PLANAR MAPS

Just as SLE is the "canonical" random non-self intersecting path, Liouville quantum gravity (introduced by Polyakov about 30 years ago in the context of string theory) is the "canonical" random two-dimensional Riemannian surface.

Just as there are discrete ways to create random paths (by drawing them on grids) there are also discrete ways to produce random surfaces (by gluing together identical unit squares along their edges). A fundamental question is whether (and in what sense) the discrete models converge to the continuum models as one constructs them using increasingly large numbers of (increasingly small) squares.

• 4. YANG-MILLS GAUGE THEORY AS SOME KIND OF CONTINUUM "RANDOM CONNECTION"

A "connection" on a lattice can be viewed as a rule assigning a Lie group element to each directed edge of the lattice. By composition, this also assigns a group element to each path and in particular to each closed cycle. We can say that two of these lattice collections are equivalent up to gauge transformation if they assign the same group element to each closed cycle.

Given the lattice and Lie group, there is a particularly natural way to choose a "random" connection modulo gauge transformation. A big problem is to make sense of a continuum limit as the mesh gets finer, or at least to suggest some kind of analogous continuum "random connection" that should be the limit. The Yang-Mills problem proposed by the Clay Institute is to construct a continuum random object in some persuasively canonical way and to show that it has a particular property (the so-called "mass gap").