9 problems

Let $R_\Lambda:\mathcal{K}\to\mathcal{K}$ be the renormalization-group map, let $\beta(k)$ be its beta function, and let $k^*$ be an RG fixed point satisfying $\beta(k^*)=0$. Let…

Let $\mathcal{K}$ be the space of kernels, let $\mathcal{S}^*(k)$ be the optimized path entropy for a kernel held fixed at $k$, and let a self-consistent kernel $k^*$ be a fixed po…

Let $\mathsf{P}_{s,\beta}$ denote a one-dimensional non-singular Riesz gas with homogeneity parameter $s$, and let $\mathsf{Leb}_{\mathbb{R}}$ be Lebesgue measure on $\mathbb{R}$.…

Let $\beta=(\beta_1,\ldots,\beta_p)$ have product prior $\pi_0=\pi^{\otimes p}$, where $\pi$ is a probability distribution on $[-1,1]$. Let $\mathcal{Z}_p$ denote the linear-regres…

Consider the model on a regular tree of degree parameter $d$, with broadcast parameter $\theta$, and a BMS survey channel $W$. Write $P_e(W)$ for the survey error probability, and…

Let $\bar\Omega$ be the limiting region with red boundary segments corresponding to constrained boundary conditions and blue segments corresponding to free boundary conditions. Let…

Let ${\mathbb Z}^d_n$ be the even discrete torus with vertex set $[n]^d$, where $n$ is even, and nearest-neighbor edges taken modulo $n$. Let…

Let ${\mathbb Z}^d$ be the nearest-neighbor graph on the $d$-dimensional integer lattice. For an integer $q\geq 3$, consider the $q$-coloring model, whose Gibbs measures are infini…

Let $M_L$ be the Markov matrix of the asymmetric annihilation process on a system of size $L$. The characteristic polynomial has only $2L$ distinct eigenvalues out of a possible…