Sample questions

Theory Questions: Example 1

In the fully connected depinning model discussed in the lectures, avalanches occur when an unstable site jumps to the next pinning well and redistributes stress to all other sites.

Assume that the distribution of distances to instability is $P(x)=2e^{-2x},x>0,$ and that the jump sizes are distributed as $g(\mathrm{\Delta })=e^{-\mathrm{\Delta }},\mathrm{\Delta }>0.$

Which of the following statements is correct?

• A. Avalanches are subcritical because the average kick is smaller than the average gap.
• B. Avalanches are critical because the average kick equals the average gap.
• C. Avalanches are supercritical because the average kick is larger than the average gap.
• D. The information provided is insufficient to determine the regime.

Theory Questions: Example 2

Consider the spherical $p$-spin model in the $T<T_c$ spin-glass phase, where $T_c$ is the temperature at which Replica Symmetry Breaking occurs. Consider the overlap $q(\sigma ,{\sigma }^{\prime })$ between two configurations extracted with the Boltzmann measure: is this a self-averaging quantity? Why?