# An inequality for a "generalised random energy model"

Let, for all $$i, j$$, $$Z_{i,j}$$ be a standard normal, chosen iid. For each $$n\geq 1, k\geq2$$, define the Hamiltonian $$H_{n,k}: [k]^n \to \mathbb{R}$$ by $$(j_1,j_2,\ldots,j_n) \mapsto \sum_{i=1}^n Z_{i, j_i}$$

So we have $$n$$ sites, each of which can be in one of $$k$$ different spins, and the energy of a configuration is just the sum of the energies over the sites. I think of this as a toy model for a more complicated system where we have interactions -- sort of like how the random energy model is a toy model for a system with $$\pm1$$ spins.

This induces a Boltzmann measure on $$[k]^n$$ in the standard way for every inverse temperature $$\beta\geq0$$. Let $$J = (J_1,J_2,\ldots,J_n)$$ be a random sample from this measure (Note that there is some notational ambiguity here -- we of course actually have one sample for each triple $$(\beta,n,k)$$, but this will be clear in context), so that $$\mathbb{P}(J=(j_1,j_2,\ldots,j_n)) \propto e^{\beta \sum_{i=1}^n Z_{i, j_i}}$$. Denote expectations with respect to this random Boltzmann measure by $$\langle\cdot\rangle_{\beta,n,k}$$, and expectations with respect to the disorder of the system by $$\mathbb{E}[\cdot]$$.

Let $$e_{n,k} = \max_{(j_1,\ldots,j_n)\in[k]^n}\sum_{i=1}^n Z_{i, j_i} = \max_{(j_1,\ldots,j_n)\in[k]^n}\sum_{i=1}^n H_{n,k}((j_1,\ldots,j_n))$$. It is completely obvious that for every realization of the $$Z_{i,j}$$ and all $$n, k$$, $$e_{n,k+1}\geq e_{n,k}$$ -- we are just taking the maximum over a larger set, so of course the maximum cannot decrease. Likewise it is easy to see that $$\mathbb{E}[e_{n+1,k}] \geq \mathbb{E}[e_{n,k}]$$, since increasing $$n$$ by one adds the maximum of $$k$$ standard normals, whose expectation is positive.

So at $$\beta = \infty$$ the picture is very clear. My question is what happens to this at finite $$\beta$$? Does it hold that $$\mathbb{E}\left[\left\langle H_{n,k+1}(J)\right\rangle_{\beta,n,k+1}\right] \geq \mathbb{E}\left[\left\langle H_{n,k}(J)\right\rangle_{\beta,n,k}\right]?$$

It turns out to be surprisingly hard to get some intuition for what should be going on in this model at finite temperatures, at least for me. I think we cannot hope for the above inequality without the expectations over the disorder, since otherwise the $$Z_{i,k+1}$$ could all be chosen to be bad -- but I can't make precise to myself what "bad" should mean.

More importantly, is there some clean technique for showing such an inequality? It feels like it should be nearly obvious if it is true, since the zero-temperature case is so trivial, but I've tried doing an interpolation argument for just the case $$n=1$$ and it became really messy.

It is worth noting that of course under this Boltzmann measure the $$j_i$$ are independent, so the effect of increasing $$n$$ is just to increase concentration of the resulting quantities.
So, here are two simulations of what is going on in the case of $$n=1$$ with $$k$$ growing,at $$\beta=5$$ and $$\beta=1$$ respectively. There are four lines in each plot -- the black, red, and blue lines are $$\left\langle H_{1,k}(J)\right\rangle_{\beta,1,k}$$ computed exactly for three different realizations of the $$Z_{i,j}$$ (the same realizations, with the same colouring, across both plots), while the purple line is the mean over $$1000$$ realizations of this quantity (with different realizations for different $$k$$).
Hopefully these plots are useful for intuition for others as well -- it seems like it is, at $$\beta=5$$, for each specific realization, making jumps and then slowly decreasing, while on average it is increasing (so the jumps are probable enough to outweigh the decay). For $$\beta=1$$ the decay is faster and the jumps less pronounced, but the overall picture could still be the same. But perhaps the purple line is actually converging, not slowly increasing?