Mathematics Problem Archive

Let $G$ be a planar graph circle-packed in $\mathbb{R}^2$ and let $f:V(G)\to\mathbb{Z}$ change by at most one across every edge. Add from each vertex ...

In the circle-packed planar-doubling setting of Section 7, prove that the expected distance of simple random walk from its root at time $t$ is at most...

Let $G$ be a planar triangulation with uniform volume growth faster than quadratic. Must $p_c(G)<1$? More strongly, is $p_c(G)=1/2$?...

For every infinite vertex-transitive graph $G$, prove that Bernoulli percolation has no infinite cluster at criticality, i.e. $\theta_G(p_c(G))=0$....

Let $X$ be a finite-energy, translation-invariant percolation process on $\mathbb{Z}^2$ satisfying the FKG inequality. If $X$ percolates almost surely...

Let $H_k$ be a $k$-regular planar triangulation. At the critical percolation parameter for the event that an open cluster contains a full infinite bin...

Does there exist an automorphism-invariant finite-energy percolation subgraph $X$ of $\mathbb{Z}^d$ that percolates almost surely but whose own Bernou...

Prove that Bernoulli percolation has at most one infinite cluster on every bounded-degree graph roughly isometric to $\mathbb{Z}^d$....

For every infinite vertex-transitive graph $G$, prove that $p_c(G)<p_u(G)$ if and only if $h(G)>0$....

For bounded-degree graphs, prove that the property $p_c<p_u$ is invariant under rough isometry....

For which $p$ does there exist $C<\infty$ such that, for any vertices $x,y$ and any $z$ on a geodesic from $x$ to $y$, $$\mathbb{P}_p(x\leftrightarrow...

Prove that a transient simple branching random walk on any vertex-transitive graph has infinitely many ends....

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...

Prove that no infinite connected Ramanujan graph is Liouville; equivalently, every such graph admits a nonconstant bounded harmonic function....

Prove that every bounded-degree graph roughly isometric to a nonamenable Cayley graph is non-Liouville....

Suppose $\mathbb{Z}$ acts on a graph $G$ by isometries, the quotient $H=G/\mathbb{Z}$ is Liouville, and simple random walk on $G$ visits every transla...

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 ...

Let $(G_k)$ be a locally convergent sequence of bounded-degree graphs, each having separation profile of order at most the square root of the subgraph...

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...

Prove that there is a universal constant $C$ such that every finite connected vertex-transitive graph $G$ of degree $d$ satisfies $$R_{\mathrm{eff}}(u...

Among all finite connected vertex-transitive graphs rescaled by their diameters, which one minimizes Gromov–Hausdorff distance to the round sphere $S^...

Does there exist a family $(G_n)$ of finite $d$-regular graphs with $|G_n|\to\infty$ and a constant $h>0$ such that every induced metric ball in every...

Let the Euclidean plane or hyperbolic plane have a triangulation whose triangles have diameter at most $r$. Suppose that for every pair of radius-$r$ ...

Characterize the graphs that occur as tangency, or nerve, graphs of sphere packings with disjoint interiors in $\mathbb{R}^d$....

Prove that every sphere packing in $\mathbb{R}^3$ whose tangency graph is $\mathbb{Z}^3$ has at most one accumulation point in the one-point compactif...

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 ...