Extension to stretched-exponential disorder tails

Establish that the universal annealed-complexity lower bounds for Gaussian-divisible spherical spin glasses remain valid when the centered non-Gaussian disorder component satisfies the weaker tail bound \(\nu(\{y:|y|\ge x\})\le \beta e^{-\alpha x^{\gamma}}\) for some \(\alpha,\beta,\gamma>0\), rather than an exponential tail bound.

Background

The main theorems assume that the non-Gaussian component of the Gaussian-divisible coupling has subexponential tails in the specific form $\nu(\{|y|\ge x\})\le \beta e^{-\alpha x}$. The authors state that they expect the results to persist under the weaker stretched-exponential assumption with exponent $\gamma>0$.

Most parts of the proof are described as extendable after modifying the truncation procedure, but the density estimate requires a new tilt involving truncated random variables whose truncation depends on the spin degree $p$.