(Carol) Xia Hua
Office hours: Monday and Wednesday 11-12Math 18.177 covers a series of topics motivated by statistical mechanics, conformal field theory, and string theory. Many of these topics arise naturally and independently in mathematics (e.g., in complex analysis, combinatorics, geometry, and the study of Brownian motion). No physics background is required for the course. The course will have four main components:
UNIFORM SPANNING TREE TOPICS (from the online book by Lyons and Peres): reversible Markov chains, stationary distribution, harmonic functions on graphs, maximum principle solution to the Dirichlet problem as the expected value at the time the processes hits the boundary (proved on graphs, stated for Brownian motion in the plane), electrical networks, probabilistic interpretations of effective resistance, star space and cycle space, effective resistance as the minimum energy of a unit flow, Rayleigh monotonicity, Kirchhoff's theorem relating edge probabilities in the UST to the unit current flow, Foster's theorem as a corollary, negative correlations of edges in the UST (a consequence of Kirchhoff's thm and Rayleigh), Burton-Pemantle theorem expressing edge marginals as determinants (note: single-marginal case is Kirchoff's theorem, two-edge case gives negative correlations), Wilson's algorithm, and cycle popping
PERCOLATION TOPICS (from Percolation by Grimmett Chapters 1,2,5,7,8,11; or alternatively, Percolation by Bollobas and Riordan ): existence of critical percolation probability p_c in (0,1), general upper and lower bounds on p_c via Peierl's argument, FKG and BK inequalities, Russo's formula, Burton-Keane argument for uniqueness of infinite cluster, continuity of percolation probability (except possibly at critical point, for d>2), Zhang argument for ruling out coexistence of infinite cluster and infinite dual cluster in two dimensions, exponential decay of cluster size for p below p_c, Kesten's theorem: p_c = 1/2 for 2d bond percolation, renormalization based proof of the continuity of critical percolation probability on half spaces, convergence of the critical percolation probability on the slabs Z^2 x [0,n] to the critical percolation probability on Z^3.
PROBLEM SET TWO (DUE MARCH 18)
ISING MODEL TOPICS (see Gibbs Measures and Phase Transitions by Georgii ): stochastic domination, FKG, existence of stochastically maximal and minimal Gibbs measures mu_+ and mu_-, translation invariance of mu_+ and mu_-, subsequential limits and compactness, uniqueness of infinite cluster when one exists (Burton-Keane argument), no co-existence of infinite cluster and dual cluster on 2D triangular lattice (Zhang argument), non-equivalence of mu_+ and mu_- and exponential decay for finite cluster sizes at sufficiently low temperature (Peierl's argument).
SLE TOPICS (from Werner's St. Flour Lecture Notes ; see Lawler's book on the subject for a more detailed account; to learn about stochastic calculus and Ito's formula, begin with Lalley's online lecture on the subject and proceed to the more detailed accounts in the books on stochastic calculus by Karatzas and Shreve or Oksendal or Revuz and Yor ): Loewner evolution and growing families of hulls, half plane capacity, definition of radial and chordal SLE, characterization of SLE via domain Markov property, Ito's formula, explicit computation of interval hitting probability, Bessel processes of arbitrarily real dimension d, criticality of d=2 Bessel process for transience, criticality of kappa = 4 for SLE hitting boundary, SLE(2) and SLE(8) as scaling limits of LERW and UST, SLE driving function change under change of domain, locality property for kappa = 6, Cardy's formula and percolation scaling limit, restriction property for SLE(8/3), reflected Brownian motion, local equivalence of boundary of Brownian excursion and boundary of Brownian motion and boundary of SLE(6).
GAUSSIAN FREE FIELD TOPICS (see Gaussian free fields for mathematicians and the introductions here and here): discrete and continuum Gaussian free field definitions, Gaussian Hilbert spaces, variance and covariance formulae, conformal invariance, Markov properties and expectations, couplings of Gaussian free field with forward and backward SLE, Liouville quantum gravity and the KPZ formula, random planar maps and scaling limit conjectures.