Mathematics Problem Archive

Give quantitative estimates improving the known bound on $\ell^*(\Delta)$, the infimum edge intensity among SIRSNs with mean unit-distance route lengt...

Show, perhaps under regularity hypotheses on a SIRSN, that for $2<\beta<4$ the paper's source-destination measure with displacement density $|z|^{-\be...

Does the major-road subnetwork $E(\infty,1)$ of a SIRSN almost surely have exactly one unbounded connected component?...

Under what additional assumptions, if any, is $\mathbb{E}\sup_{i\ge1}\operatorname{len}[R(0,U_i)]<\infty$ for independent uniform points $U_i$ in the ...

For $k$ uniform random points $Z_1,\ldots,Z_k$ in a square of area $k$, prove $\mathbb{E}\operatorname{len}[\operatorname{span}(Z_1,\ldots,Z_k)]\sim\e...

Prove that in a scale-invariant random spatial network the subnetwork $S(1)$ cannot be a tree, even allowing Steiner points....

For the complete graph with i.i.d. uniform edge weights revealed online, prove that the minimum expected cost $\mathbb{E}Y_n$ of an online spanning-tr...

Give a reasonably explicit description of the unique stationary distribution of the one-dimensional Hammersley-type process whose particles drift righ...

For the limiting maximum-flow density $v(p)$ in oriented bond percolation on the square lattice, prove $1-v(p)\sim\sqrt{2(1-p)}$ as $p\uparrow1$....

Let $X$ and $X'$ be discrete-time stochastic processes on the same state space, and let $\theta_n$ denote the shift. If $\mathbb{P}(\theta_nX\in A)\to...

If setwise convergence $\mathbb{P}(\theta_nX\in A)\to\mathbb{P}(X'\in A)$ for every measurable path-space set $A$ does not imply total-variation conve...

Suppose $\theta_nX$ converges in distribution to $X'$ on a separable metric path space. Is there a coupling characterization involving only a joint co...

Call an infinite connected graph $G$ of uniformly bounded degree an infinite expander if there is a constant $c>0$ such that, for every vertex set $S$...

In Gromov's density model with generators $a,a^{-1},b,b^{-1}$, choose $3^{nd}$ relators independently and uniformly from the reduced words of length $...

Call a finite subset $X\subset S^2$ symmetric if a finite group acts transitively on $X$ by isometries. Determine an explicit universal upper bound fo...

Does there exist a graph $G$ and a map $f:V(G)\to\mathbb{R}^2$ such that $|\|f(x)-f(y)\|_2-d_G(x,y)|<C$ for every $x,y\in V(G)$ and some constant $C<\...

Must every vertex-transitive graph of exponential growth contain an infinite subgraph $H$ with positive Cheeger constant $h(H)>0$? Can $H$ always be c...

Must every graph with uniform exponential volume growth contain an infinite subgraph, possibly a tree, having positive Cheeger constant?...

Does there exist a vertex-transitive graph whose multiplicative rough-isometry constants to its $k$-net graphs tend to $1$ as $k\to\infty$?...

Does there exist a graph for which repeatedly passing to $k$-net graphs, at appropriately chosen scales, produces strictly smaller large-scale graph m...

Let $G$ be vertex-transitive and let $G_k$ be a $k$-net graph of $G$. Must $h(G_k)\ge h(G)$ whenever $G_k$ is not a single vertex?...

There is a positive function $f(h,d,k)$ such that every graph $G$ with $h(G)>h>0$ and maximum degree less than $d$ has $h(G_k)>f(h,d,k)$ for each $k$-...

Start with a unit square and repeatedly choose a current square uniformly and subdivide it into four squares. Let $D_n$ be the minimum number of curre...

Must every Cayley graph of superlinear growth contain, up to rough isometric embedding, either the square grid $\mathbb{Z}^2$ or an infinite binary tr...

If an infinite graph is $C$-roughly transitive for some finite $C$, must it be roughly isometric to a homogeneous metric space? Equivalently, does the...

For every Cayley graph $G$, does there exist $r=r(G)$ such that $G$ covers every graph whose radius-$r$ balls are all isomorphic to the radius-$r$ bal...

Fix a rooted radius-$r$ ball $B(o,r)$ that occurs as every radius-$r$ ball of some finite graph. What is the minimum diameter of a finite graph all of...

Let $G$ be an $n$-vertex graph whose rooted balls of size $k$ are all isomorphic. If $k>n/2$, or if $k$ is within a fixed constant of $\operatorname{d...

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 $\mu(G)>1$....

If an infinite vertex-transitive graph is $f(r)$-sofic for some function $f$, must it be $cr$-sofic for a constant $c$? More uniformly, for fixed degr...

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 ...

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...

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...