Statistical Mechanics and Lattice Models

Spring 2020
Wednesdays 1-3:50pm
Burkle 22

Instructor: Professor Allon Percus

Contact information: CGU Math South, tel. 909-607-0744

Office hours: Wednesdays 10-11 at CGU Math South, or by appointment.

Course description: An intermediate-level graduate course emphasizing fundamental techniques in mathematical physics. Topics include: random walks, lattice models, asymptotic/thermodynamic limit, critical phenomena, transfer matrix, duality, polymer model, mean field, variational method, renormalization group, finite-size scaling.

Background and prerequisites: Math 251 or equivalent course in probability. The course will assume basic familiarity with thermodynamics, as well as undergraduate-level real analysis and linear algebra.

Textbook: All required materials, including course notes, will be posted on Canvas under the “Files” section.

Assignments and grading: Homework will generally be assigned weekly, and will be due the following week. There will be a take-home final exam. Grading will be approximately as follows:

  • 50% Homework
  • 40% Final exam
  • 10% Class participation

You may work together on homework assignments, but all your answers must be explained in your own words. Your homework submission must also state the names of all those with whom you collaborated. Please familiarize yourselves with CGU’s academic honesty policies.

Tentative schedule:

  • Jan 22: Probabilistic background. Central limit theorem; large deviations.
  • Jan 29: Random walks. Master equation; first-passage probability; critical dimension.
  • Feb 5: Statistical mechanics models. Ising and related models; equilibrium properties.
  • Feb 12: Phase transitions. Thermodynamic limit, first- and second-order transitions; universality.
  • Feb 26: Exact solutions. Transfer matrix; Onsager solution; spherical model.
  • Mar 4: Rigorous approaches. Griffiths’ inequalities; Peierls argument.
  • Mar 11: Duality. High-temperature expansion; low-temperature expansion; duality.
  • Mar 18: SPRING BREAK
  • Mar 25/Apr 1: Mean field methods. Variational approach; complete graph; trees.
  • Apr 8/Apr 15: Fundamentals of renormalization group. One-parameter renormalization; two-parameter renormalization.
  • Apr 15/Apr 22: Real-space renormalization group. Majority rule; decimation.
  • Apr 29: Monte Carlo methods. Random number generation; importance sampling; Metropolis method; Finite-size scaling.
  • May 6: Statistical mechanics of computation. Satisfiability; phase transitions; algorithmic consequences.