310 problems

Let $d\ge 1$. Suppose $D(x)$ has finite support $\mathcal{U}$. If $D(x)$ is uniformly distributed on $\mathcal{U}$, and if every $x\in \mathcal{U}$ is on the boundary of…

Let $G$) be a connected $d$-regular graph with adjacency matrix $A$ and $n$ vertices. Let $\xi_1,\xi_2,\ldots,\xi_n$ be orthogonal eigenvectors of $A$, with the conditions stated b…

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.

Let the Laplacian operator be the operator defined by the jump conditions in --, and let $h_1$ denote the function in Theorem. Uniqueness conjecture. The unique, up to multiplicati…

Let $n$ particles evolve according to the Attracting Random Walks (ARW) model on an arbitrary graph, with interaction parameter $\beta<0$. The mixing time is measured with respect…

The shell-based inpainting Gaussian-blur conjecture. If $u_0$ is independent of its $y$-coordinate, then

Let $(T_n(k))_{k \geq 0}$ denote the flip walk on triangulations of the sphere with $n$ vertices, and let its mixing time be measured with respect to the uniform distribution on su…

A high-dimensional random walk on a discrete point process is a random walk on such a point process in sufficiently large dimension. Failure conjecture. There are random walks on d…

A two-dimensional random walk on a discrete point process is a random walk in dimension $2$ whose state space is a discrete point process. Existence conjecture. There are transient…

Let ${\cal LF}_n$ be a locally free group, and consider its random walk with word length $K$ and drift $l$, the limiting expected normalized word length. Let $T(w)$ be the roof of…

Let ${\cal LF}_n^+$ be the locally free semigroup (the heap) and ${\cal LF}_n$ the locally free group (the colored heap). For a random walk word $w_N$, let $T(w_N)$ denote its roof…

Let $(A,u)$ be a finitely generated quantum group, and let $b_k$ and $p_k$ denote the growth and random-walk quantities constructed in the paper. Here $f_k\approx g_k$ means that t…

Let $d\geq 3$, let $\mathcal{S}(1)=\{\mathbf{z}\in\mathcal{Z}_d:\|\mathbf{z}\|=1\}$, and define the occupation time of the unit-ball surface centered at $\mathbf{z}$ by … Writing…

Consider a graph obtained by a non-trivial stitching of the lattices $\mathbb{Z}^{d}$ and $2\mathbb{Z}^{d}$. Stitching isotropy conjecture. Any such graph is an isotropic graph. If…

Long-range percolation scaling conjecture. If $\alpha\in(0,2)$, then the corresponding random walk scales to a symmetric $\alpha$-stable Lévy process in $\mathbb R^d$.

High-dimensional corrector tightness conjecture. Let $d\gg1$. Then for each $\epsilon>0$ there exists $K<\infty$ such that

Theorem 5.1 conjecture. Theorem 5.1 is true in all $d\ge2$.

Let $N_\phi$ be the 3-manifold obtained from a random Heegaard splitting determined by a mapping class $\phi$, and let $L$ denote the random-walk length. The hyperbolicity and line…

Let $d$ be fixed. For a finitely supported function $\xi:{\mathbb Z}^d\to{\mathbb R}$, let $W_{2n}({\bf v},\xi)$ denote the associated lattice random-walk quantity at ${\bf v}$ and…

Four-dimensional logarithmic-correction conjecture. The accurate upper-bound scale for the length of $L$ is

Critical-dimension conjecture. The critical dimension is

Loop-erased random walk lower-bound conjecture. The expected length of the loop-erasure of $R$ satisfies

Recovery conjecture. The bias function $\eta$ can be recovered up to a shift only.

Let $p\in(p_c,1)$, where $p_c=\frac{1}{2}$ is the critical probability for bond percolation on $\mathbb{Z}^2$. For the biased random walk on the infinite percolation cluster, let…

Let $G$ be a black box group and let $N\lhd G$. The Andrews--Curtis graph $\Delta_k(G,N)$ supports a nearest-neighbour random walk, and the AC-replacement algorithm $ACR_k(G,N)$ ru…