47 problems

Tightness conjecture. The highest points of sufficiently long positive arches, and the lowest points of sufficiently long negative arches, are tight to the Brownian limit.

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…

Fix $z\in[0,1]$ and let $(u_0,v_0)$ be the corresponding point on the Logan--Shepp--Vershik--Kerov limit curve $\Omega_*$. Let $w_1,w_2,\ldots$ be i.i.d. uniform random variables o…

Let $U$ and $R$ be domains in the setting of Theorem, let $z$ be the specified starting point, and write $\tau_U$, $\tau_R$, $\tau_{U^+}$, and $\tau_{R^+}$ for the corresponding fi…

Let the c3-finite bead process be conditioned to survive up to distance $r$, equivalently up to time $r^2$ or imaginary part $r$, and let $r\rightarrow\infty$. Scaling limit conjec…

Consider the Brownian Pool model obtained by replacing the continuous-time and continuous-space random walks in the Pool model by Brownian motion, and let $\lambda=1$ be the critic…

Brownian motion and cutoff conjecture. For every $t\geq 0$, the state $\varphi_t$ is not absolutely continuous with respect to the Haar state. Moreover,

For each $N\geq 1$, let $T_{2N}$ and $T_{2N+1}$ be the absorption times in the discrete Whittaker process. Let $M_N$ be the maximum height of $N$ non-intersecting reflected Brownia…

Let $K\subset\mathbb{R}$ be a nonrandom compact set, let $\mathfrak S(\theta)$ be the random set of points of slow growth at parameter $\theta$, and let $\lambda(\theta)$ denote th…

Let $\theta_c$ be the critical parameter and let $\mathfrak S(thetac)$ denote the corresponding random set of points of slow growth. Write $\operatorname{\dim{{rm H}}}$ for Hausdorf…

Let $W$ be the Brownian motion and let $G$ be the Green kernel appearing in the definition of the renormalised Amperean area. Write $I_W$ for the recentred self-intersection local…

Let $B$ be a one-dimensional Brownian motion on $[0,\infty)$ and let $K$ be its almost surely existing concave majorant on $[0,\infty)$. Ouaki–Pitman conjecture. The process $2K-B$ h…

General-potential conditioning conjecture. As $T\to\infty$, the probability measures

Let $B$ be $n$-dimensional Brownian motion. High-dimensional Brownian motion conjecture. For all sufficiently large $n$, $B$ is not minimal almost surely; moreover, for all suffici…

Bessel coupling conjecture. There exists a coupling of $(R,Y)$ and $\widetilde{\mathbf{B}}$ such that almost surely there are $\delta_1>0$ and $n_1\in\mathbb{N}$ for which, for eve…

Existence and uniqueness conjecture. Existence and pathwise uniqueness for the solutions of these SDEs hold for every

Biased-sign construction conjecture. For $\rho=1$ and every $p\in[0,1]$, the solutions $\bm{\mathscr Z}_{1,1-p}^{(u)}$ can be constructed using independent biased signs satisfying

Let $n$ particles move in $d$ dimensions, sampled at intervals of length $\delta$, and let $\widehat{\pi}_k$ be the estimator at time step $k$. Let $T_{\max}$ denote the maximum tr…

Assume that the Lévy process $X$ has finite lifetime $\zeta\in(0,\infty)$. Let … be the last time at which the Euclidean norm is maximal. Set $M:=X_\tau$ if…

Let $B=(B_t)_{t\geq 0}$ be a standard one-dimensional Brownian motion, and let $K=(K_t)_{t\geq 0}$ be its almost surely unique concave majorant on $[0,\infty)$. Write $R_5$ for a f…

Let $\{B_i(\cdot)\}_{i\in\mathbb{Z}}$ be a sequence of two-sided standard Brownian motions on $\mathbb{R}$, and let $\Pi_n$ be the almost surely unique geodesic from $(0,0)$ to…

Quenched scaling conjecture. For almost every such $\rho$,

For positive integers $A,B,C$, let $P_{A,B,C}(\sigma)$ denote the first hitting probability of the permutation $\sigma\in S_3$ in the gambler's ruin walk, and let…

Unit-cube conjecture.

Let $\{\{\tilde{X}_{\alpha,t}\}_{t}\}_{\alpha\in\mathbb{R}}$ be a grand coupling of Brownian motions $\{\mathrm{BM}(\alpha)\}_{\alpha\in\mathbb{R}}$. A failure probability bound is…