# Simplification through the replica symmetry Ansatz

I'm going through some lecture notes dealing with the replica method and I feel like that I did not fully understand the concept of the replica symmetry (RS) Ansatz. At some point in the notes we come to an integral looking like this (I simplified it a little bit for this question): $$\prod_{\alpha=1}^{n}\left(\int dm^{(\alpha)}\right)\exp\left[-\sum_{\alpha=1}^{n}m^{(\alpha)}\right]$$ where $$\alpha$$ is my index for all the $$n$$-replicas I have.

To simplify this expression we now use RS, saying that $$m^{(\alpha)} = m$$. The notes say that using this RS we essentially get: $$\int dm\exp\left[-nm\right]$$

The thing that is confusing me now is why we are left with only one integral. Naively I would have thought we would get something like this: $$\left(\int dm\right)^n\exp\left[-nm\right]$$ but I guess this is wrong and the reason that I'm wrong is probably some misunderstanding on what the RS Ansatz really is. That's the reason I wanted to try and ask here if anyone could help me and give me a better understanding on this topic. Thank you!

The key point is precisely that, once you assume that the integral is dominated by replica-symmetric configurations, the variables $$m^{(\alpha)}$$ on each replica are no longer independent; in particular, when performing the integral, you just need one single representative $$m$$. If you want to wrap a few equations around this just to make it slightly more explicit, you can imagine that the replica symmetry ansatz amounts to replacing the integral
$$\prod_{\alpha=1}^n (\int dm^{(\alpha)}) \,\exp\left[-\sum_{\alpha=1}^nm^{(\alpha)}\right]$$
$$\int dm \exp(-m)\times \prod_{\alpha=2}^n (\int dm^{(\alpha)})\delta(m^{(\alpha)}-m) \,\exp\left[-\sum_{\alpha=2}^nm^{(\alpha)}\right]$$ where I chose the first replica as the representative. Then, the result follows immediately.
• Oh wow, thank you! That's a really nice explanation in my opinion. Maybe a short follow up question to this: I wrote down your expression for the simple case of $n=2$ replicas and this thought came to my mind: Before doing the RS Ansatz, $m^1$ and $m^2$ could be any value. But after doing the Ansatz I constrain their value to be on this $45$ degree diagonal between them (if draw a graph with $m^1$ on the $x$ and $m^2$ on the $y$ axis). And this constraining on the diagonal is what happens with this delta distribution. Would this be a valid way to talk about this? Jul 11, 2023 at 15:27