Statistical Mechanics
Statistical mechanics studies lattice models of interacting spins and bonds through Gibbs measures, partition functions, correlation inequalities, and phase transitions.
Most of the subject follows the Ising model from a finite graph to the infinite lattice. On a finite graph the Lee–Yang circle theorem puts the zeros of the ferromagnetic partition function, as a polynomial in the fugacity \(e^{-2\beta h}\) of the external field \(h\), on the unit circle, and the Griffiths and Griffiths–Hurst–Sherman inequalities control how spins correlate. On the lattice, the Peierls argument gives long-range order at low temperature, the ordered half of the phase transition.
The two-dimensional model is exactly solvable. Kramers–Wannier duality exchanges high and low temperature. Kac–Ward matrices and Sherman's loop cancellations relate Kac–Ward determinants to sums over closed walks, the spin Kac–Ward identities express the partition function on the torus through four such determinants, and Onsager's solution computes the free energy per site in the infinite-volume limit.
The largest field is the random-cluster model. The Edwards–Sokal coupling couples Potts and Ising spins with random subgraphs, turning spin correlations into connection probabilities, and from measures on finite graphs the field builds infinite-volume measures, the free energy and the critical point. Random currents and the switching lemma lead to the sharpness of the phase transition for the Ising model.
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