Background: the Parisi formula gives an exact expression for the free energy of the SK model. The formula (at least the upper-bound) can be derived by looking at the free energy, and then replacing the "actual" objects involved in computing it (i.e. the spins and the Gibbs measure) with objects from a "Ruelle Probability Cascade" (i.e. the leaves and the weights of a tree, given by some explicit formulas involving a Poisson process).
In particular, the energy of a leaf on this tree (which is supposed to represent $H(\sigma)$ for each $\sigma$) is given by the product of some Poisson random variables.
Questions: consider the SK model at low temperature for large-but-finite $n$, and the associated Ruelle Probability Cascade.
- How should I think of the vertices at each depth of the tree? I know they have some fixed $L^2$ norm; do they have some interpretation as elements of $[-1, 1]^n$?
- In particular, do they correspond to vectors in $\{-1,0,1\}$, so that descending down the tree is like revealing more coordinates? This would be very nice, because then the weights associated with vertices at each depth would be something like the change in energy when you reveal some more coordinates.
- If this, or something similar, is the case, why would the increments in energy be Poisson rather than Gaussian? It seems to me that
(I know this question is not self-contained, but I assume only someone very familiar with the SK model would be able to answer this, and thus would be familiar with these concepts anyway. Let me know if anything is unclear and needs expanding / a more formal definition)
