Math 8450, Fall 2014, Topics in Mathematical Physics

Proposed reading projects.

Project 1:

Concerning the use of Ising models for approximating lattice phi-4 models, read the article "The (phi4)2 field theory as a classical Ising model" by B. Simon and R. B. Griffiths, Comm. Math. Phys. 33 (1973), 145--164.

Project 2:

If you liked the transfer matrix diagonalization for the 1d Ising model and would like to know how this works in 2d, read "Two-dimensional Ising model as a soluble problem of many fermions" by T. D. Schultz, D. C. Mattis, and E. H. Lieb, Rev. Modern Phys. 36 (1964), 856--871.

Other related articles which can also be helpful:

- "Algebraic derivation of the partition function of a two-dimensional Ising model" by C. J. Thompson, J. Math. Phys. 6 (1965) 1392--1395. Can be used as an introduction to the SML article.

- "Odd operators and spinor algebras in lattice statistics: n-point functions for the rectangular Ising model" by D. B. Abraham, Comm. Math. Phys. 59 (1978), no. 1, 17--34. It makes connections to an important method called the Bethe Ansatz.

- "Discrete holomorphicity and Ising model operator formalism" by C. Hongler, K. Kytölä and A. Zahabi, to appear in "Analysis, Complex Geometry, and Mathematical Physics: A Conference in Honor of Duong H. Phong", (Contemp. Math. Series of AMS). It makes connections to recent developments regarding the conformal invariance of the 2d Ising scaling limit.

Project 3:

Read the original article on the GHS inequality, namely, "Concavity of magnetization of an Ising ferromagnet in a positive external field" by , R. B. Griffiths, C. A. Hurst and S. Sherman.

Project 4:

Regarding the Lee-Yang Theorem, read "Some applications of the Lee-Yang theorem" by J. Fröhlich and P.-F. Rodrigez, J. Math. Phys. 53 (2012), no. 9, 095218, 15 pp. It might be useful to also look at "Characterization of Lee-Yang polynomials" by D. Ruelle, Ann. of Math. (2) 171 (2010), no. 1, 589--603.

Project 5:

The most powerful method regarding correlation inequalities is Aizenman's random current representation which builds on the switching lemma from the GHS paper of Project 3. To learn more, read "Signs of the Ising model Ursell functions" by S. Shlosman, Comm. Math. Phys. 102 (1986), no. 4, 679--686.

Other related articles which can also be helpful:

- "Random currents and continuity of Ising model's spontaneous magnetization" by M. Aizenman, H. Duminil-Copin and V. Sidoravicius, to appear in Comm. Math. Phys. More recent reference, good as an entry point.

- "Geometric analysis of phi4 fields and Ising models. I, II" by M. Aizenman, Comm. Math. Phys. 86 (1982), no. 1, 1--4. The original article on random currents.

Project 6:

For a good and rather self-contained treatment of phi-4 type lattice models, read the MIT Ph. D. Thesis "Continuous-Spin Ising Ferromagnets" by G. S. Sylvester.

Project 7:

For a recent development on scaling limits and representation theory, read "Lattice representations of the Virasoro algebra I: discrete Gaussian free field" by C. Hongler, F. Johansson Viklund and K. Kytölä.

Project 8:

For important examples of Gaussian measures on the space of temperate distributions, read the two reviews "Fractional Gaussian fields: a survey" by A. Lodhia, S. Sheffield, X. Sun and S. S. Watson; and "Log-correlated Gaussian fields: an overview" by B. Duplantier, R. Rhodes, S. Sheffield and V. Vargas.

Project 9:

For the treatment of the "plus" boundary condition via the FKG inequality and the DLR formalism for infinite volume Gibbs measures, read Chapters 3 and 6 of the book "Equilibrium Statistical Mechanics of Classical Lattice Systems: a Concrete Introduction" by S. Friedli and Y. Velenik.

Project 10:

Regarding the interplay of topology and statistical mechanics, read "Topology and phase transitions I. Preliminary results" by R. Franzosi, M. Pettini and L. Spinelli; and "Topology and phase transitions II. Theorem on a necessary relation" by R. Franzosi and M. Pettini.


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