### 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 19: PROBLEM SESSION
- 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.