59 problems

Random-memory threshold conjecture. The same threshold governs the case when $\{D_n\}_{n \ge 1}$ is random.

Glynn–Whitt constant conjecture. Based on simulations, they conjectured that $c=2$. This conjecture concerns the explicit value of the constant in the Brownian-percolation asymptot…

The bindweed model is studied on a rooted tree with random-environment transition matrices, and its behavior is classified by a parameter such as the critical value associated with…

Let $(X_n)_{n\ge0}$ be the biased random walk on the rooted tree described in the paper, let $|X_n|$ denote its distance from the root, and suppose that $\lambda>1$. Logarithmic ma…

Mytnik–Xiong's persistence conjecture. The process $X$ will converge weakly, as $t\to\infty$, to some non-trivial random measure $X_\infty$ on ${\mathbb R}^d$.

Consider the setting of Theorems --, with the convergence rates established there. Optimality conjecture. The convergence rates obtained therein are optimal up to slowly varying fa…

Let $m(0)$ denote the odometer at the origin for fixed-energy activated random walk with i.i.d. initial particle density $\rho$, and let $\rho_c(\lambda)$ be its critical density f…

For a directed polymer in a random environment, let $\beta_c$ denote the critical point separating weak and strong disorder, and let $\bar\beta_c$ denote the critical point separat…

Let $X(t)$ denote the position of the random walker at time $t$, and let $d$ be the dimension. Write $f(t)\ll g(t)$ when $f(t)$ is bounded above by a constant multiple of $g(t)$, a…

Uniform-integrability conjecture. Under this condition, the family $\{\tilde Z_N\}_{N\in\mathbb N}$ is uniformly integrable and converges weakly to an $L^1$-integrable limit.

Let $H\in(\frac12,1)$ be the correlation exponent, let $\alpha$ be the tail exponent of the renewal process, and let $\tilde Z_{N,\beta_N}^{\omega,h_N}$ denote the rescaled Wick-or…

Let $\mathcal{C}_p$ be the class of probability distributions on $\Omega$ that are independent and identically distributed over the vertices, isotropic, and have expected degree eq…

Consider a branching random walk in random environment on $\mathbb Z^d$: particles move as continuous-time simple random walks and branch at position $x$ at rate $\xi_x$, where…

Consider the dynamic exclusion-process environment with particle jump rate $\nu>0$, and let $v(\rho)$ denote the speed of the random walk at density $\rho$. A zero-speed interval m…

Let $L$ be an irreducible dispersal matrix, let $A_i=R_i+L$ be the matrices in the differential inclusion model, and let $x_i$ be eigenvectors of $A_i$. Assume that the eigenvector…

Let an RWCE be a random walk in a changing environment, and call it non-adaptive when the environment's evolution is independent of the walk's history. The source discusses recurre…

Central limit conjecture. Theorem 2 and Corollary 2.2 should both hold in dimensions $d=3,4$ after replacing $T^{-1/2}$ by $T^{-3/4}$ when $d=3$ and by $T^{-1/2}(log T)^{-1}$ when…

Let $Z(n,m)$ denote the point-to-point log-Gamma polymer partition function, and call a process $(h(k))_{k\geq0}$ horizontal-stationary when the solution with initial data…

Consider the half-space log-Gamma polymer with bulk weights distributed as $\operatorname{Gamma}^{-1}(2\theta)$ and diagonal weights as $\operatorname{Gamma}^{-1}(\alpha+\theta)$,…

Threshold conjecture. There exists $\kappa_0\in(-1,0)$ such that the displayed assumption holds in this setting if and only if $\kappa>\kappa_0$.

Shape-function linearity conjecture. For $\kappa$ sufficiently close to $-1$, the shape function $\Lambda$ does not have a linear piece.

Let $(Z_n)_{n\geq 0}$ be the random walk defined in Section 2, and let $\mathbb{P}$ denote its probability measure. Law of large numbers conjecture. There exists…

Simplified-model local limit conjecture. For all sufficiently large $n$, for the upper extremal point $M_n^+$ and every integer $k$,

Local limit conjecture. Almost surely,

Asymptotic shape conjecture. There exists a bounded convex set $U\subset\mathds{R}^d$ such that, for every $\varepsilon>0$,