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Let $G=(V,E)$ be a finite connected graph with $n$ nodes. Let $P$ denote the transition matrix representing Glauber dynamics for the Ising model on $G$., and fix a time $t$. Which two initial configurations $x,y$ maximize the total variation distance $\|P^t(x,\cdot)-P^t(y,\cdot)\|$?

The obvious guess is that $x,y$ should be the all $+1$ and all $-1$ states, respectively. The challenge is to prove (or disprove) this guess.

REFERENCES
[1] Markov Chains and Mixing Times