32 problems

Consider the Plancherel-TASEP interacting particle system, and let $u_n$ be the position of its second-class particle at time $n$. Let $\mu_{\operatorname{SC}}$ be the semicircle d…

Let $c1_{2,3}(s,n)$ denote the quantity appearing in the preceding conjecture, with $s$ fixed and $n$ varying. The recurrence conjecture. … The source explicitly says that, using t…

Let $c1_{2,3}(s,n)$ denote the quantity counting the relevant continuous multiline-queue configurations, let $N$ be the total size parameter used in the paper, and let $f_{(s,s)}$…

Let $b3^\ell(s,n)$ denote the number of continuous multiline queues with the specified one-wrapping configuration, and let $f_{(n-1,s+1)}$ denote the relevant hook-length count. Th…

Let $\rho ameq{}(0,1)$ be the stationary Bernoulli density, let $\mathbb{P}\rho$ be the corresponding TASEP path measure, and let $ht(j)$ be the height function. For $winmathbb{R}$, d…

Let $\rho_-$ and $\rho_+$ be the left and right Bernoulli densities for the TASEP, let $\bar h(y)$ be the hydrodynamic limiting height profile, and let $h_t(j)$ be the height at ti…

Consider a finite multi-floor TASEP with $c\ge 1$ floors and attempted particle arrival and departure rates $\alpha>0$ and $\beta>0$. Let … and let … Assume condition (a): $L^*$ co…

Let an impurity in the totally asymmetric simple exclusion process have jumping rates $b1$ and $b2$, and initially be located at the interface of a $1$-$0$ profile. Assume that…

Let $K_1$ be the product of the first three matrices in the displayed factorization of the TASEP kernel, and let $K_2$ be the operator defined by … Here the operators act on the in…

Convergence conjecture. For $\alpha>\frac12$, the half-line open TASEP with step initial condition $\eta_t^{\mathrm{step}}$ converges to the probability measure…

In the setting of the section, consider right Bernoulli TASEP with sequential updates and an additional caterpillar of length $b\varepsilon^{-1/2}$ placed at distance…

Consider TASEP with a reservoir of rate $\lambda=\rho q$ for some $\rho \in (0,\infty)$, and suppose that a flow rule holds for a flow of strength $q$. Let $\pi_{\lambda}$ be the e…

Let $\rho0: \mathbb{R}\rightarrow[0,1]$ be an initial density profile, and let ${nu^{\epsilon}}{epsiloninmathbb{R}{0}}$ be probability measures on the configuration space…

Let $\rho$ be the particle density, let $h$ be the critical density, let $\phi(\rho;n)$ be the finite-$n$ typical flux, and let $\phi^*$ be the maximal-equilibrium-state flux. Let…

Let $\rho$ be the particle density, let $h$ be the critical density, and let $\phi^*$ denote the maximal-equilibrium-state flux. Consider TASEP-H with I-perturbations. High-density…

Let $n$ be a non-negative integer. For independent standard Brownian bridges $b_0,\ldots,b_{2n}$, define … Here $b$ is a standard Brownian bridge whose average is represented by…

Let $n$ be a non-negative integer. For independent standard Brownian bridges $b_0,\ldots,b_{2n}$, define … Here $\max$ denotes the maximum over the bridge parameter. Let…

Let $n$ be a non-negative integer and let $h_0$ be an arbitrary sufficiently regular continuous function satisfying … For independent standard Brownian bridges $b_1,\ldots,b_n$, de…

Let $\boldsymbol\tau$ and $\boldsymbol\nu$ be the model parameters, let $\psi_{\bf w}(\boldsymbol\tau,\boldsymbol\nu)$ denote the component associated with a particle configuration…

For a continuous TASEP on a ring with $n$ particles, let … be the stationary two-point correlation for an adjacent pair, where $a$ is arbitrary. The upper-right correlation conject…

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 $N\ge n$, let $\mathbf{k}=(k_1\ge\cdots\ge k_r\ge 1)$ be a partition, and let $\mathbf{k}'$ be its conjugate. For $S\subseteq [r]$, let $\mathbf{k}(S)$ be the partition consist…

Let $N\ge n>k\ge 1$ and let $0\le b_1<\dots<b_n\le N-1$. For the permutation $s_kw_0$ obtained by applying the simple reflection $s_k$ to the longest permutation $w_0$, let $A_k$ b…

Let $w=(w_a)$ denote the cyclic multispecies TASEP word, and let $E_{i_1,\dots,i_r}$ and $E_{j_{r+1},\dots,j_{r+s}}$ denote the corresponding nearest-neighbour block probabilities.…