7 problems

A spatial random permutation (SRP) is a random bijection of a finite lattice or set of sites; the Mallows model is one SRP whose graph has a concentrated band structure around the…

Main conjecture. Under the spectral-gap and Lipschitz-eigenfunction assumptions, the limit defining the normalized partition function holds with the Fredholm determinant…

Limiting Fredholm determinant conjecture. Under suitable assumptions, $C$ is the Fredholm determinant of a certain integral operator related to the Schrödinger bridge, the optimal…

Suppose that $n(1-q)^2\to\beta\in(0,\infty)$ and $s/n\to\alpha$. Let $X$ be the exponential random variable conjectured to describe the limiting distribution of…

In the microscopic regime $n(1-q)^2\to\infty$, let $\mathcal{C}_s$ be the cycle containing $s$. The normalized extrema are $(1-q)^2(\max(\mathcal{C}_s)-s)$ and…

Consider a Mallows permutation with parameter $q$ and the cycle $\mathcal{C}_s$ containing $s$, in the microscopic regime $n(1-q)^2\to\infty$. Let $x\ge 0$. Exponential diameter-ta…

Let $\pi_n\in S_n$ be a Mallows random permutation with parameter $q(n)$ satisfying $n(1-q(n))\rightarrow\beta$, and let $\mu_{\rho_\beta}$ be the permuton induced by the density ……