# Week 3 - The Replica Symmetric Estimate of SK Free Energy

Antares Chen University of Chicago

### Abstract

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.

Date
Feb 15, 2022 10:30 AM — 12:00 PM
Event
Algorithmic Methods in Statistical Physics
Location
JCL 390

### Recap of the Meeting

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

$$\mathbb{E} \big[ Z_n^r \big] = \sum_{\{ \sigma^{(a)}i \}a} \prod{i,j}^n \sqrt{\frac{n}{2\pi}} \int e^{- \frac{n}{2} \cdot z{ij}^2} \cdot \exp \bigg( \beta \sum_{a=1}^r z_{ij} \cdot \sigma_i^{(a)} \sigma_j^{(a)} \bigg) \, dz_{ij}$$

### Material

• [pdf] Scanned presentation notes