# Where is simple exponential relaxation in relation to graph theory?

I'm studying relaxation in the Ising model. Let me describe my model from a physics perspective. Let's say we have $$n$$ spins in one dimension, e.g. $${\hat s_o=(\uparrow,\downarrow,... \text{n times}....\uparrow)}$$. I'm going to leave out interactions because they aren't relevant to my question. With each increment of time $$t$$, I will flip one randomly selected spin. Let $$\hat s(t)$$ denote the state of the system at time $$t$$, then let me define the relaxation as $$\hat s_o \cdot \hat s(t)/n$$. My physics intuition tells me that in the limit of a large system this relaxation behaves like simple exponential decay $$\text{exp}(-t/n)$$, give or take factors of $$2$$. Indeed, you can easily calculate this relaxation numerically with a computer, and you can achieve simple exponential relaxation to any desired precision by choosing a large enough $$n$$.

Next, I'll to switch to a math perspective. In math, the non-interacting Ising model with single spin flip dynamics is equivalent to a random walk on a hypercube of dimension $$n$$. I'm new to graph theory, but I have been studying mixing times in a simple graph and I've made it as far as Cheegers inequality.

To my question: Why don't I see a calculation of simple exponential relaxation on a hypercube in graph theory?

In a physics approach, simple exponential relaxation in a noninteracting Ising model is a simple, obvious result. Why isn't it an obvious result in the field of graph theory? I can't seem to find the bridge between the two approaches.

Thanks.

Update: I think I'm getting close to an answer. There is a paper by https://www.researchgate.net/publication/235197003_Spectral_Graph_Theory_of_the_Hypercube which studies the eigenvalues of the $$n$$ hypercube. I think that the relaxation of the Ising model corresponds to the fastest Eigenvalue and this holds in the limit of large $$n$$.

• Let $X_k$ be the expected number of spin that have not change sign at time $k$. You could write down the recursive formula between $X_{k+1}$ and $X_k$. This lead to a equation which should give a geometric decay (and hence exponential decay). Commented Apr 4, 2023 at 0:59