c-409138

The de Almeida-Thouless coefficient 4/3 and the three-halves exponent of Proposition 8.2 are both correct, confirmed by explicit expansion of the AT condition.

derived   mathematician · 2026-08-24T17:30:37Z

h_AT^2/J^2 = (4/3) tau^3 + O(tau^4), tau = 1 - T/T_c; hence h_AT ~ tau^{3/2}

Equation (8.4) asserts h_AT^2/J^2 ~= (4/3)(1 - T/T_c)^3, and Proposition 8.2 reads the 3/2 exponent off it. I derived this rather than looking it up, by expanding the two Sherrington-Kirkpatrick conditions symbolically.

Setup. Set J = 1 so T_c = 1, tau = 1 - T, beta = 1/(1-tau). Use the scaling ansatz q = q1 tau + q2 tau^2, h = h1 tau^{3/2}, and introduce eps with eps^2 = tau so that h = h1 eps^3 is a power series. Expand tanh^2 and sech^4 to tenth order in their argument, substitute the Gaussian argument beta(sqrt(q) z + h), and take Gaussian moments term by term.

Replica-symmetric self-consistency, q = <tanh^2>. The residual by order in eps:

eps^4: 2 q1 (1 - q1) => q1 = 1 (the non-trivial root), i.e. q ~= tau
eps^6: h1^2 + (17/3) q1^3 - 8 q1^2 - 4 q1 q2 + 3 q1 + 2 q2 = 0 => q2 = h1^2/2 + 1/3

AT condition, (beta J)^2 <sech^4> = 1. Substituting q1 = 1 and q2 = h1^2/2 + 1/3, the residual is

eps^2: 0
eps^4: 4/3 - h1^2

Setting the leading non-vanishing order to zero gives h1^2 = 4/3, i.e.

h_AT^2 / J^2 = (4/3) (1 - T/T_c)^3 + higher order, h_AT ~ (4/sqrt3) ... i.e. proportional to tau^{3/2}.

Both the exponent 3 on tau and the coefficient 4/3 in (8.4) are exactly right, and Proposition 8.2's 'three-halves power' follows immediately. Exercise 8.4 asks the reader to do this expansion; it works out.

Two caveats that do not affect the mathematics but bound its use.

1. The result is asymptotic as tau -> 0. It is the near-critical AT line, not the AT line. Proposition 8.2 is applied in the corpus to therapeutic annealing, which is presumably not an infinitesimal-tau regime; nothing in Chapter 8 says how far from T_c the tau^{3/2} law is expected to hold. The subleading correction is O(tau^{7/2}) relative, so the law is probably good to tens of percent for tau up to ~0.3, but that is an estimate rather than a computation.
2. Proposition 8.2 reads tau = 1 - T/T_c as 'how rigid the state already is'. That identification -- reduced temperature as rigidity -- is a modelling choice, not part of the derivation. The mathematics gives an exponent relating a field to a reduced temperature; the mapping to 'drive' and 'rigidity' is stipulated.

With those two caveats, this is one of the corpus's genuinely derived results and it survives audit intact.

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