62 problems

Let $G=K_n$ be the complete graph on $n$ vertices. Independently assign each edge weight $n^{1+gamma}$ with probability $1/n$ and weight $1$ otherwise. Let $mathbb T$ be the set of…

Let $\varphi:\mathbb{Z}^d\to\mathbb{Z}$ be an integer-valued minimal surface in linear disorder, with disorder strength $lambda$. Roughening-transition conjecture. For sufficiently…

Let $K$ be the population-size scaling parameter, let $\hat{X}^K$ denote the rescaled epidemic process, and let $\Lambda<0$ be the non-persistence parameter. Consider initial condi…

Consider the Matheron–de Marsily model, a random walk $(M_n)_{n\in\mathbb N}$ in a randomly oriented lattice on $\mathbb Z^2$, and let its survival probability be the probability t…

Let $d\geq 2$, and let ${\mathtt P}_{\omega}[T(0,1)>n]$ denote the quenched probability that the branching random walk has not reached the relevant target by time $n$ in environmen…

The Menshikov–Petritis conjecture. The additional conditions needed for the chaos problem to have nontrivial solutions should be the same as those needed for the random walk to be…

Let $M$ denote the maximum position of a subcritical branching random walk in a random environment, and consider the cases $alpha>1$ with $varTheta(1)>0$ or $alpha\leqslant 1$. Und…

Let $G=K_n$ be the complete graph, with independent edge weights taking the value $n^{1+\beta}$ with probability $1/n$ and the value $1$ otherwise. Let $\beta$ vary with $n$, and l…

Equality of the tail constants. The constants $C_1$ and $C_2$ are equal. This would sharpen the established two-sided asymptotic-order estimate for the tail probability of the whol…

Let $X$ be a random walk in Dirichlet environment on ${\mathbb Z}^2$ with positive parameters $(\alpha_1,\ldots,\alpha_4)$. Define … A local accelerating function $\gamma$ is a pos…

Let $X$ be a random walk in Dirichlet environment, and let $\mathbf{(T')}$ denote Sznitman's condition in a unit direction $\hat u$: directional transience means … and the renewal-…

Let $X$ be a random walk in Dirichlet environment on ${\mathbb Z}^2$ with drift vector $d_\alpha=\sum_{i=1}^4\alpha_i e_i$. Redundancy conjecture. The recurrence hypothesis and the…

Let $X$ be a random walk in Dirichlet environment on ${\mathbb Z}^2$ with drift vector $d_\alpha=\sum_{i=1}^4\alpha_i e_i$. A walk is directionally transient in a unit direction…

Recurrence conjecture. If $d_\alpha=0$, then the RWDE is recurrent.

Kolmogorov–Spitzer conjecture. The normalized passage times $n^{-1/k}T_n$ could converge to a stable limit law.

Let $Lambda_\infty=\mathbb{Z}^d$ be the infinite domain, with dimension $d$ and codimension $n$. A minimal surface on $Lambda_\infty$ is an infinite-domain minimal surface in the m…

Let $Lambda_L=\{-L,\ldots,L\}^d$ with $1\leq d\leq 3$, let $eta=\eta^{\mathrm{white}}$, and consider the regime $lambda_0\leq\lambda\leq 1$, where $lambda_0=L^{-\frac{4-d}{2}}$. Le…

Let $Lambda_L=\{-L,\ldots,L\}^d$ with $1\leq d\leq 3$, and let the disorder be $eta=\eta^{\mathrm{white}}$. Write $xi$ for the transversal fluctuation exponent in dimension $d$. Di…

Consider the long-range contact process (LoRaC) on the spatial model of the paper, with edge length $|y-x|$ and a parameter $\alpha>0$. Fewer edges. The LoRaC has a phase transitio…

Let $q^{(t)},p\in(0,1)$ be given and let $\alpha>1$. Let $q^{(x)}$ denote the parameter controlling the sparse spatial environment in the long-range contact process (LoRaC). Extinc…

Let $q^{(t)},q^{(x)},p\in(0,1)$ be given, and let $\alpha>1$. The long-range contact process (LoRaC) is the percolation model described in the paper, with edges whose length distri…

Fisher's scaling conjecture. The annealed density has the scaling form

Let $\mathcal{T}^{\boldsymbol{\mu}}$ be a branching process in varying environment, and let … where $(\overline{\mu_i})_{i\geq 1}$ is the associated sequence of mean offspring para…

Let $P$ denote the law of the random environment for a ballistic random walk in random environment on $\mathbb{Z}^2$. An invariant measure is equivalent to $P$ when it has the same…

Conjecture. Necessary and sufficient conditions for recurrence and transience of $(M_n)_{n\geqslant 0}$ in terms of the random parameter $(A,B)$ are difficult to come by.