Abstract
A disordered system is denominated `annealed' when the interactions themselves may evolve and adjust their values to lower the free energy. The opposite (`quenched') situation when disorder is fixed, is the one relevant for physical spin-glasses, and has received vastly more attention. Other problems however are more natural in the annealed situation: in this work we discuss examples where annealed averages are interesting, in the context of matrix models. We first discuss how in practice, when system and disorder adapt together, annealed systems develop `planted' solutions spontaneously, as the ones found in the study of inference problems. In the second part, we study the probability distribution of elements of a matrix derived from a rotationally invariant (not necessarily Gaussian) ensemble, a problem that maps into the annealed average of a spin glass model.
Highlights
IntroductionConsider the following problem: we are given a system of size N depending on disorder variables J, and a set of distributions PJ , that we expect to depend exponentially in N
Our study of the annealed average of spin-glass models shows that the freedom of the couplings to adapt to the spin configurations leads at low temperatures to self-planted solutions [6]
In [28] we showed that the probability distribution of a sub-block of size 2 × 2 inherits the property of rotational invariance of the original ensemble, namely that
Summary
Consider the following problem: we are given a system of size N depending on disorder variables J, and a set of distributions PJ , that we expect to depend exponentially in N. Finite-dimensional systems with short-range interactions, one can imagine the system as being composed of many quasi-independent parts, and conclude that the free-energy, energy and entropy (but not their exponentials) are the addition of their values in these parts This argument suggests that these are the quantities that have to be averaged, since they are the ones that concentrate in probability in the observed value, their average that gives the typical results for one sample. Our study of the annealed average of spin-glass models shows that the freedom of the couplings to adapt to the spin configurations leads at low temperatures to self-planted solutions [6] These are configurations with particular low energy with respect to the quenched ones as a result of the annealing. In the second part (section 3) we proceed to calculate the (annealed) joint distribution of an r × r submatrix (r finite) of a large N × N random matrix derived from a general rotationally invariant matrix model
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