Mathematics Problem Archive

If $G$ is strongly amenable, can Bernoulli percolation on $G\times\mathbb{Z}$ have infinitely many infinite clusters throughout a nondegenerate interv...

Let $G$ be an infinite graph with $p_c(G)=1$. For Bernoulli percolation on $G\times\mathbb{Z}$, must every infinite cluster intersect each fiber $\{v\...

Let $G$ be a bounded-degree expander and suppose some vertex $v$ satisfies $$\mathbb{P}_{1/2}\!\left(\operatorname{diam}(K_v)>\tfrac12\operatorname{di...

In one-dimensional long-range percolation with edge probabilities proportional to $\beta|i-j|^{-2}$, study the distance exponent $\theta(\beta)$ defin...

Can two simple random walks on $\mathbb{Z}^3$ or $\mathbb{Z}^4$, started at vertices at graph distance $10$, be coupled so that their paths are disjoi...

Let $G$ be the one-skeleton of a bounded-degree triangulation of an open disk. If $G$ is transient, prove that Bernoulli site percolation with paramet...

Tile the unit square by finitely or countably many squares of varying sizes, with at most three squares meeting at a corner, and color the squares ind...

Let $G$ be a bounded-degree triangulation of an open disk with polynomial volume growth. Prove that its Bernoulli site-percolation critical probabilit...

Let $G$ be the one-skeleton of a bounded-degree recurrent triangulation of an open disk. Prove that Bernoulli site percolation with parameter $1/2$ ha...

Let $G$ be the one-skeleton of a bounded-degree transient triangulation of an open disk. Prove that Bernoulli site percolation with parameter $1/2$ ha...

In the Poincaré disk, sample a Poisson process of intensity $\lambda$ with respect to hyperbolic area, form its Voronoi tessellation, and color cells ...

Construct the Poisson–Voronoi tessellation of the plane equipped with an $\ell_p$ metric and give the cells their adjacency graph metric. What is the ...

Delete each edge of the square lattice independently with probability $q<1/2$, condition the origin to lie in the infinite component, and let $K_q$ be...

Let $D_n$ be the source-to-sink first-passage distance in the recursively substituted hierarchical graph whose distances satisfy $D_n\stackrel d=D_{n-...

For the hierarchical first-passage distances $D_n$ satisfying $D_n\stackrel d=D_{n-1}+\min(D'_{n-1},D''_{n-1})$, determine concentration around the me...

On the three-branch hierarchical graph $G_n$ described in Section 9.3, launch external-DLA particles from the sink until a particle settles at the sin...

In the random hierarchical graph obtained by repeatedly replacing a uniformly chosen edge by the fixed three-edge pattern of Section 9.4, let $D_n$ be...

Start from one edge and at each stage replace every edge independently by two edges in series with probability $p$ or two edges in parallel with proba...

Does there exist a mixing random tiling of the Euclidean plane whose law is invariant under rotations and translations, is stationary under a local cl...

For which dimensions $d$ can $\mathbb{R}^d$ be partitioned into pairwise disjoint curves, each of which has the law or geometric regularity of a Brown...

For i.i.d. first-passage percolation on $\mathbb{Z}^2$, prove or disprove that boundary fluctuations have a Tracy–Widom limit and that the variance of...

Prove that natural i.i.d. first-passage-percolation models on $\mathbb{Z}^d$, including exponential edge lengths, almost surely contain no two-sided i...

Run two walks with a common clock on $\mathbb{Z}^d$, each choosing uniformly among neighbors not previously visited by the other walk. Prove that in $...

Take a uniform quadrangulation with $N$ faces conditioned to have genus $CN$, where $0<C<1/4$. Prove that its rooted local limit is the stochastic hyp...

Determine the almost-sure asymptotic growth rate of the effective resistance from the root to graph-distance $r$ in the uniform infinite planar triang...

Let $G$ be a distributional local limit of finite planar graphs. Prove that $p_c^{\mathrm{site}}(G)\ge1/2$ almost surely and that there is no infinite...

On the $n\times n$ grid with a Gaussian free field with no boundary conditions, give every vertex length equal to the exponential of the field. If $\g...

Generate a quadrangulation from $2n$ bits using the Schaeffer bijection and independently resample each bit with probability $\varepsilon$. Determine ...

For every $p\ge4$, determine the joint law of the matrix of pairwise distances among $p$ independent points sampled from the volume measure of the Bro...

Let $G$ be a bounded-degree recurrent planar triangulation with a fixed root. Are there arbitrarily large $r$ and finite domains containing the radius...

Let $(G_n)$ be finite transitive graphs with $|G_n|\to\infty$ and uniformly bounded effective resistances between all vertex pairs. Prove that the lar...

For every $C<\infty$, prove that there is $c=c(C)<1$ such that, for every simple $n$-vertex graph $G$, the probability that simple random walk covers ...

Let $G$ be a bounded-degree triangulation of a disk. Prove that each of the following conditions implies $p_c(G)\le1/2$: $\operatorname{Dim}(G)\ge2$; ...

Let $G$ be a connected quasi-transitive graph and let $p\in(0,1)$. If Bernoulli percolation has more than one infinite cluster almost surely, prove th...

Give general conditions implying $p_u(G)<1$. In particular, prove or disprove that every one-ended transitive graph has $p_u(G)<1$....

Let $G$ be a planar graph and consider Bernoulli percolation at $p=1/2$. If an infinite open cluster exists almost surely, prove that there are almost...

For a quasi-transitive graph $G$, characterize when Bernoulli percolation has a unique infinite cluster at $p=p_u(G)$. In particular, give necessary a...

For nearest-neighbor independent bond percolation on $\mathbb{Z}^d$, $d\ge2$, let $p_c(d)$ be the critical edge-retention probability. Prove that at $...

On $\mathbb{Z}^d$, start with the origin black and every other vertex white. Repeatedly choose uniformly an edge having one black and one white endpoi...

Let $(X_n)_{n\in\mathbb{Z}}$ be a centered strictly stationary sequence with $\mathbb{E}[X_0^2]<\infty$. For $k\ge1$, define $$\phi_X(k)=\sup_m\sup\bi...

Start with $n$ tetrahedra and glue their faces together at random. The vertex links need not be spheres but are essentially random triangulated surfac...