Develop rigorous theory for quenched averages with non-Gaussian (sum-of-squares) cost functions

Develop a rigorous mathematical framework to evaluate quenched averages of logarithmic partition functions for models where the cost function is a sum of squared Gaussian terms, as in the constrained random least-squares problem considered here, going beyond the cases with normally distributed costs where rigorous methods already exist.

Background

Swapping disorder and thermal integrations in quenched averages is delicate. Rigorous results exist for models with normally distributed cost functions, but the cost function in this problem is a sum of squared Gaussian terms, which is not normally distributed.

The author explicitly notes that for this non-Gaussian sum-of-squares structure, a rigorous theory has not yet been developed. Establishing such a theory would close a key gap and could provide a rigorous foundation for the results currently obtained via heuristic replica methods.