Random Energy Model — Spin Glass Transition

Derrida REM, energy level statistics, freezing at T_c, replica symmetry breaking

Energy Level Distribution

12
1.0
REM (Derrida 1980): 2^N configurations, each with independent random energy E_α ~ N(0, NJ²/2).
Ground state: E_min ≈ −J√(N ln2). Energy density histogram is Gaussian.

Free Energy vs Temperature

1.0
Glass transition at T_c = J/√(2 ln 2).
High-T (liquid): f(T) = J²/(4T) (annealed average).
Low-T (glass): f(T) = −J√(ln2) (frozen, dominated by ground state). Entropy = 0 below T_c (frozen in one state). This is 1-step RSB.

Glass Phase — Boltzmann Weights

1.50
12
Above T_c: Boltzmann weights spread over many configurations (ergodic).
Below T_c: weights concentrate on O(1) lowest-energy states — system "freezes" into a glass. Replica symmetry breaks: q_EA = ⟨m²⟩ ≠ 0.

Replica Symmetry Breaking (q distribution)

0.80
Order parameter: P(q) = distribution of overlaps between replicas.
Paramagnetic (T>T_c): P(q) = δ(q). Glass (T<T_c): P(q) = (1-m)δ(q) + m·δ(q-q_EA). 1-step RSB in REM. Full RSB (Parisi) in SK model.

Extreme Value Statistics — Ground State Energy vs N

Minimum of 2^N Gaussians: E_min ≈ −J√(N ln2)·√2 (follows from Gumbel extreme value theory). This sets the critical temperature via competition between entropy (2^N states) and energy (lowest state).