5 problems

Let $m=d+1$ equiprobable signals of equal energy be transmitted as points in $\mathbb R^d$ through an additive white Gaussian noise channel, and let the receiver use maximum-likeli…

Let $q\geq 3$, $r\in \\{1,\dots,q-1\\}$, and consider the fuzzy Potts model on the torus $\mathbb{T}^d$. Denote by $J_c$ the critical value corresponding to non-uniqueness of Gibbs…

Let $J>0$, $q\geq 3$, and $r\in \\{1,\dots,q-1\\}$. Consider the fuzzy Potts model on the torus $\mathbb{T}^d$, and let $\gamma_p$ be Bernoulli product measure of density $p$. Defi…

Let $q\geq 3$, $r\in \\{1,\dots,q-1\\}$, and consider the fuzzy Potts model on $\mathbb{Z}^d$. For $J>0$, write $\nu_{q,J,r}^{\mathbb{Z}^d,+}$ for its plus-state fuzzy Potts measur…

Let $\mu\in\mathfrak{P}_n$ be a probability measure on subsets of $[n]$, and for $0\leq k\leq n$ let $\mu_k$ be its conditional measure given that the sampled set has size $k$. Wri…