the Ising and Potts models
/ Ising: EE-zing; Potts: POTS /
The Ising model is the canonical mathematical model of a ferromagnet and the most-studied system in statistical mechanics. It asks: if neighbouring atomic spins prefer to align, when does a macroscopic magnet spontaneously magnetize? It is the cleanest setting in which a phase transition — the order-disorder transition at a critical temperature — can be both physically motivated and rigorously analyzed, and it sits next to percolation through a deep equivalence (the random-cluster representation).
On a finite box of Z^d place a spin sigma_x in {-1, +1} at each site. The energy (Hamiltonian) of a configuration sigma is H(sigma) = - sum over neighbouring pairs {x,y} of sigma_x sigma_y (minus an external-field term h sum_x sigma_x), so aligned neighbours lower the energy. The Gibbs/Boltzmann measure weights each configuration by e^(-beta H(sigma)) / Z, where beta = 1/(kT) is the inverse temperature and Z = sum over sigma of e^(-beta H) is the partition function (the normalizing constant whose derivatives generate all thermodynamic quantities). At high temperature (small beta) the spins are nearly independent and the system is disordered; at low temperature (large beta) the alignment term dominates and the system can develop long-range order. The spontaneous magnetization m*(beta) is the limiting average spin per site in the infinite volume with plus boundary conditions and zero field; it is zero above the critical inverse temperature beta_c and strictly positive below it, signalling the transition. The Potts model generalizes this to q colours instead of two spins: each site takes a value in {1,...,q} and neighbouring equal colours lower the energy; q = 2 recovers Ising. In d = 2 the Ising critical point is exactly solvable (Onsager, 1944), with magnetization m* = (1 - sinh^(-4)(2 beta))^(1/8) below criticality.
Why this matters: the Ising model is the proving ground for nearly every idea in rigorous statistical mechanics — correlation inequalities (GKS, FKG, Lebowitz), the Peierls argument for a phase transition in d >= 2, the absence of a transition in d = 1, exact solution in d = 2, and conformal invariance of the 2D critical model (Smirnov again, with SLE_3 interfaces). The honest subtleties: there is NO phase transition in dimension 1 (a single misaligned bond costs only finite energy, so order is destroyed at any positive temperature), and the existence of multiple Gibbs measures (the +/- states) below beta_c is precisely the rigorous meaning of 'spontaneous symmetry breaking'. The Potts model with large q exhibits a FIRST-order (discontinuous) transition, unlike Ising's continuous one, so 'the' nature of the transition depends on q and d.
On Z^2 at low temperature, start the Ising model in a box with all boundary spins +1. Below beta_c the bulk magnetization stays positive even as the box grows: the plus boundary 'wins' and a net magnetization survives in the infinite-volume limit. Above beta_c (high temperature) the boundary's influence dies away and the bulk magnetization is 0 regardless of boundary condition — the two boundary conditions give the SAME Gibbs state.
Boundary conditions matter below beta_c (multiple Gibbs states) but not above it (a unique state).
There is NO phase transition in dimension 1 — a single domain wall costs only finite energy, so any positive temperature destroys long-range order. And the transition's type depends on the model: Ising is continuous, but the q-state Potts model with large q is first-order (discontinuous).