Antares Chen University of Chicago
We begin by introducing the limit identity that drives the replica method calcuation, and give an overview of how the computations are carried out. We then compute the action integral form of the free energy, and give an overview of the family of Replica Symmetry Breaking ansatzen. We conclude by computing the replica symmetric free energy estimate.
We began our discussion of the replica method by discussing the replica limit identity
The following identity holds when the expectation is taken over a distribution with finite moments. $$ \mathbb{E} \big[ \log Z_n(\beta) \big] = \lim_{r \rightarrow 0} \frac{1}{r} \cdot \log \Big( \mathbb{E} \big[ Z_n^r \big] \Big) $$
The replica method is then a heuristic computation method used to estimate the free energy via the above limit identity like so $$ \lim_{n \rightarrow \infty} \frac{1}{n} \mathbb{E} \big[ \log Z_n(\beta) \big] \approx \lim_{n \rightarrow \infty} \lim_{r \rightarrow 0} \frac{1}{nr} \cdot \log \Big( \mathbb{E} \big[ Z_n^r \big] \Big) $$
We then spent the meeting on discussing
An overview to different stages of the replica computation: Pre-ansatzen, ansatzen, and post-ansatzen
Reasons why the replica method is a heuristic method of computation
The replica symmetric ansatzen, and a sketch of $k$-replica symmetry breaking
The initial steps of the computation required to deduce the action (exponential) integral form of the free energy.
We ended on an expression for $\mathbb{E} \big[ Z_n^r \big]$ given by
$$ \mathbb{E} \big[ Z_n^r \big] = \sum_{\{ \sigma^{(a)}i \}a} \prod{i,j}^n \sqrt{\frac{n}{2\pi}} \int e^{- \frac{n}{2} \cdot z{ij}^2} \cdot \exp \bigg( \beta \sum_{a=1}^r z_{ij} \cdot \sigma_i^{(a)} \sigma_j^{(a)} \bigg) \, dz_{ij} $$