13 problems

Beta-limit conjecture. As $c\to\infty$, in the two-island case both $X$ and $Y$ converge in distribution to the beta distribution with parameters $(a_1+b_1,a_2+b_2)$; in the seed-b…

Let $u$ be the transition rate from the lounge to the queue, let $\beta$ satisfy … where $\eta>0$ is constant, and let $A=\lfloor\eta\mu/\alpha\rfloor$. For each $u$, write…

Non-product-form conjecture. Then $\mathcal{G}$ has a non-product-form stationary distribution independently of the reaction rate; equivalently, it belongs to class N.

Product-form characterization conjecture. The network $(\mathcal{G},\kappa)$ has a product-form stationary distribution if and only if it is complex balanced.

Mixture classification conjecture. Under these conditions, every stationary gap distribution of an infinite system of competing Brownian particles is equal to $\pi_\rho$ for some p…

Let the diffusion coefficients satisfy $\sigma_n=1$ for every $n\in\mathbb{Z}$, and assume the conditions of Theorem 2.1 hold. For $(a,b)\in\mathbb{R}^2$, define … and let … For…

Let $g_w(q_1,\dots,q_n)$ denote the stationary probability density for the continuous TASEP configuration with permutation $w$, and let $\mathbf{k}=(k_1\ge\cdots\ge k_r\ge 1)$ sati…

Let $g_w(q_1,\dots,q_n)$ denote the stationary probability density for the continuous TASEP configuration with permutation $w$, and let $w_0$ be the longest permutation. For…

Let $\mathcal Q$ be a multiline queue with state space $\Omega^{FM}_m$, and let $\Omega_m$ be the state space of the corresponding inhomogeneous multispecies exclusion process. For…

Let the inhomogeneous multispecies exclusion process have $n$ classes of particles, with particles of class $i$ jumping at rate $x_i$. Let $w(\boldsymbol{\rho})$ denote the station…

Stronger converse conjecture. Then $\pi$ is a nonnegative measure and consequently it is a stationary distribution of $X$.

Let $S_n$ be the symmetric group and let $\zeta(w)$ be the normalized stationary-distribution component defined by the nullspace of $P^T-I$. Assume that each $\zeta(w)$ is a polyno…

Small-hole convergence conjecture. The measures $Q_r$ converge to $\operatorname{Unif}((D_1)^2)$ when $r\to0$.