1. Spin systems
Open questions proposed for spin systems, primarily the Ising and Potts models and variants therein.Recall that the Ising model on a graph $G=(V,E)$ is a distribution on $\Omega=\{\pm 1\}^{V(G)}$ with probability \[\mathbb P(\sigma)\propto \exp(\beta \sum_{(i,j)\in E} \sigma_i \sigma_j)\,. \] and the $q$state Potts model is a random assignment of colors $\sigma\in \{1,...,q\}^{V(G)}$ with probability \[\mathbb P(\sigma)\propto \exp(\beta \sum _{(i,j)\in E} \boldsymbol 1\{\sigma_i =\sigma_j\}) \]

Universal lower bound for Potts
Problem 1.1.
[Yuval Peres] Consider the $q$Potts model or uniform $q$ proper colorings on a general graph $G$. Is there a universal $\asymp n\log n$ lower bound on the mixing time uniform in the temperature? Is there always an $\frac {n\log n}2$ lower bound?
[Ding, Peres] proved an $\frac {n\log n}2$ lower bound for the Ising model on general graphs. 
Spin glass with i.i.d. couplings
Problem 1.2.
[Andrea Montanari] Consider the spin glass model on $\mathbb{Z}_n^d$ with i.i.d. couplings, i.e. with Hamiltonian given by $$ H_n(\sigma)= \sum_{(i,j)\in E_n} J_{ij}\sigma_i\sigma_j\, , $$ where $J=(J_{ij})$ is a random symmetric matrix with i.i.d. $\pm 1$ entries. The Gibbs distribution is given by $$ \mu_n(\sigma)= \frac{1}{Z_n}\exp(\beta H_n)\, . $$ Fix a realization of $J_{ij}$ and consider the Glauber dynamics. Does there exist $\beta_0$ and $\varepsilon>0$ such that, for all $\beta\geq\beta_0$, the mixing time of this chain is at least $\exp(n^\varepsilon)$? 
Fast mixing for antiferromagnetic Ising at high temperature
Problem 1.3.
[Allan Sly] Take a (random) $d$regular graph, and consider the antiferromagnetic Ising model, i.e. $J_{ij}=1$ for all $i,j$. Can we show show fast mixing $O(n\log n)$ up to $$(d1)\tanh(\beta)<1 \, ?$$ 
Mixing time for ferromagnetic Ising at critical temperature
Problem 1.4.
[Eyal Lubetzky] What is the mixing time for the ferromagnetic Ising model at critical temperature $\beta=\beta_c$ on a random $d$regular graph? Is it $n^c$? 
Censoring for the Potts model
Problem 1.5.
[Yuval Peres] Consider the $q$state Potts model (say with $q=3$ for concreteness) and start from all green. Is it the case that deterministically censoring a sequence of spin flips can only increase the total variation distance to stationarity? 
Diagnostics
We seek diagnostic tests for knowing that a general spin system is mixed on a general graph $G$.Problem 1.6.
1. Consider the ferromagnetic Ising model on a general graph $G$. Which two $x,y\in\Omega$ maximize $\P^t(x,\cdot)P^t(y,\cdot)\_{TV}$? More generally, is it true that in any monotone reversible chain, the maximal and minimal initial configurations maximize the total variation distance between two chains?
2. Can one use a diagnostic to estimate $t_{\mbox{mix}}$ up to $O(1)$? Of course, answering question (1) would answer this as well.
3. Consider the $3$Potts model for a general graph $G$. Can one devise a diagnostic to differentiate between $t_{\mbox{mix}}=O(n\log n)$ and $\exp(\Omega(n))$? 
Noisy majority model
The noisy majority model with parameter $\epsilon\in (0,1)$ is a spin system on a graph $G$ that is the stationary distribution of the following Markov chain on $\Omega=\{\pm 1\}^{V(G)}$: assign each vertex a rate$1$ Poisson clock. When the clock at a site rings, the spin at that vertex randomizes according to a $\mbox{Ber}(\frac 12)$ with probability $\epsilon$ and chooses the majority of the spins of its neighbors and itself with probability $1\epsilon$ (flipping a coin in the event of a tie).Problem 1.7.
Consider the noisy majority model on $\mathbb Z_n^d$ for $d>1$. Show there exists a fixed $\epsilon>0$ such that $t_{\mbox{mix}}\gtrsim \exp(n^{\epsilon})$ (should be true for $\exp(cn)$).
Cite this as: AimPL: Markov chain mixing times, available at http://aimpl.org/markovmixing.