851 problems

Let $\Gamma$ be a connected graph on $[n]=\{1,2,\ldots,n\}$, and identify each edge of $\Gamma$ with the corresponding transposition of $S_n$. Write $T(\Gamma)$ for the resulting s…

Let the incipient infinite cluster (IIC) be the random configuration obtained as the local limit of critical percolation conditioned on the origin being connected to distance tendi…

Let $\Gamma<\mathrm{SL}_N(\mathbb{R})$ be a discrete subgroup, and let the random walk on $\Gamma$ have finite support. Its hitting measure is a measure on the boundary at infinity…

Consider the Tree Builder Random Walk in which, at time step $n$, a new leaf is attached with probability $p_n=n^{-\gamma}$, where $\gamma$ is a real parameter. For…

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

Let $G$ be a finite graph with vertex set $V$. For a starting vertex $v\in V$, write $\mathbb{E}_v$ for expectation for the simple random walk started at $v$, and let the…

Let the once-reinforced random walk (ORRW) with reinforcement parameter $c>0$ be run on the integer lattice $\mathbb{Z}^d$. Sidoravicius's recurrence conjecture. The walk is recurr…

Aldous–Diaconis conjecture. In this regime, the random walk exhibits cutoff with high probability.

Aldous–Fill conjecture. The maximum relaxation time over connected regular graphs with $n$ vertices is

Oliveira's conjecture. The mixing time of the interchange process on $G$ should be of the same order as the mixing time of the product chain.

The two dual hypergeometric random walks are recurrent when $0<\delta\leq 1$, and there are no mass points in the corresponding measure. Null recurrence conjecture. This recurrence…

Let $\Gamma$ be a uniform lattice in $SL(2,\mathbb{R})=\operatorname{Isom}(\mathbb{H}^2)$, so that $\mathbb{H}^2/\Gamma$ is a closed orbifold, and let a random walk on $\Gamma$ hav…

Let $G=\mathbb{Z}/2\mathbb{Z}\wr\mathbb{Z}^d$ with $d\geq 1$, and let $\mu$ be a non-degenerate finitely supported measure on $G$ whose projection to $\mathbb{Z}^d$ induces a trans…

Kazhdan property (T) conjecture. In the general case, this order is conjectured to be correct; equivalently, the relevant automorphism groups of free groups should have Kazhdan's p…

Recurrence conjecture. The simple random walk on the $\nu$-IIPT is almost surely recurrent for every value of $\nu$.

Let $\Gamma$ be a finitely generated group, and let $\mu$ be a finitely supported probability measure on $\Gamma$. Suppose that the associated random walk satisfies … where $C_x$ d…

Singularity conjecture. If $\mathsf{m}$ has finite support, then the $\mathsf{m}$-stationary measure $\nu$ is singular to the Lebesgue measure class on $\operatorname{\mathcal{F}}$…

Let $G$ be a finite simple group of Lie type, let $S$ be a non-trivial conjugacy class of $G$, and let $\Gamma(G,S)$ be the Cayley graph generated by $S$. Lubotzky's conjecture. Th…

Let $d\geq 5$ and consider a discrete torus of side length $L$. For the Ising model and self-avoiding walk, let their two-point functions be defined in the usual way, and let…

Transience conjecture. Supercritical long-range percolation is transient when $d\geq 3$ and $\alpha\geq 2d$.

Critical-walk recurrence conjecture. The biased random walk $RW_{\lambda_c}$ on $T_{\mathbb H}$ or $T_{\mathbb Z^2}$ is recurrent.

Let $W_n^\beta$ denote the partition function of the time-inhomogeneous branching random walk at inverse temperature $\beta$, and let $\beta_c$ be its critical parameter. In the ti…

A probability measure on a group is non-degenerate if its support is not contained in a proper subgroup. A probability measure is Liouville when every bounded harmonic function for…

Consider a stronger-than-linear edge-reinforced random walk whose transition probabilities are proportional to a positive function $f$ of the number of edge crossings, under the hy…

Domain-of-attraction conjecture. An analogous domain-of-attraction statement should remain valid for broader mean-field graph sequences, at least when the dual random walk mixes on…