Critical Slowing Down Exponents of Mode Coupling Theory

A method is provided to compute the exponent parameter $\lambda$ yielding the dynamic exponents of critical slowing down in mode coupling theory. It is independent from the dynamic approach and based on the formulation of an effective static field theory. Expressions of $\lambda$ in terms of third order coefficients of the action expansion or, equivalently, in terms of six point cumulants are provided. Applications are reported to a number of mean-field models: with hard and soft variables and both fully connected and dilute interactions. Comparisons with existing results for the Potts glass model, the random orthogonal model, hard and soft-spin Sherrington-Kirkpatrick, and $p$-spin models are presented.

Figure 1

Rescaled ${\chi }_4(t)$ for the Viana-Bray model with $c=4$, $h=.7$, and $T=.73536(1)$, along the dAT line with $a_{\mathrm{th}}=0.406$. No fitting interpolation is performed. The line $y=x$ is parallel to the expected behavior for $t<t^{*}(N)$ (see text).