10 problems

Non-integer BP solution conjecture. The non-integer solution extends beyond the small $T_c+\varepsilon$ vicinity of $T_c$ and transitions smoothly as $T\to\infty$ into the fully ho…

Let $M$ be an $n\times n$ matrix with entries in $\{0,1\}$ and row sums $d_1,\dotsc,d_n$. Minc's conjecture. The permanent of $M$ is at most … Minc's conjecture was proved by Brégm…

Let $W$ be an $n\times n$ matrix with independent entries distributed as $N(0,1)_{\mathbb C}$, and let $p$ be a polynomial in its arguments. Permanent anti-concentration conjecture…

Let $\mathfrak S_m$ denote the permutation group on $m$ elements, let $y^i_j$ be linear coordinates on $\mathbb C^{m^2}$, and define the permanent by … A polynomial-size circuit is…

Let $A$ be a $d$-dimensional $(0,1)$-matrix of order $n$. For each $i$, let $r_i$ be the number of ones in the $i$th hyperplane of $A$. Multidimensional permanent upper-bound conje…

Let $p$ be a non-negative matrix, and let $Z_{o\text{-}f}^{(\gamma)}(p)$ denote the fractional BP partition-function estimate at parameter $\gamma$. In particular, write…

Let $\phi$ be a doubly stochastic $n\times n$ matrix, so that its entries are non-negative and every row and column sums to $1$. The permanent is denoted by…

Gurvits's BP upper-bound conjecture. For any non-negative $p$, $f(n)$ is asymptotic to $\sqrt{2}^n$. This conjecture concerns the worst-case multiplicative gap between the permanen…

Exponential critical-ratio conjecture. There is a sequence of constants $\theta_d$ with

Let $A$) be an $n\times n$ matrix with entries in $\{0,1\}$. For a matrix $A$, define $M(A)$ by … where the $\rho_{ij}$ are chosen independently from the Gaussian distribution on…