Mathematics Problem Archive

Let $G$ be an infinite bounded-degree graph. Prove that $\operatorname{I-dim}(G)>1$ implies $p_c(G)<1$. As a weaker target, prove the conclusion when ...

Does the law of loop-erased random walk on $\mathbb{Z}^d$ have the exponential intersection-tail property: for two independent sampled paths $\gamma_1...

For some $d\ge3$, is there a probability measure on embeddings of $\mathbb{Z}^2$ into $\mathbb{Z}^d$ having an analogue of the exponential intersectio...

For a subset $S=\{(n,f(n),g(n)):n\in\mathbb{N}\}\subset\mathbb{Z}^3$, characterize the conditions on $f$ and $g$ under which $S$ supports a probabilit...

Let $G$ be vertex-transitive with positive Cheeger constant. If $\mu$ is the connective constant of self-avoiding walks and $\mu_{\mathrm{loops}}$ is ...

Prove that the connective constant $\mu(G)$ is continuous under local convergence of infinite vertex-transitive graphs....

Prove that every graph $G$ with isoperimetric dimension greater than $1$ has self-avoiding-walk connective constant $\mu(G)>1$....

Let $T_n$ be a uniform triangulation of the sphere with $n$ faces, normalize its circle packing by its conformal barycenter, and let $\mu_{P T_n}$ be ...

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...

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...

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...

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...

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 $...

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

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 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...

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...