Mathematics Problem Archive

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

Prove or disprove each of the proposed implications between the SIRSN properties numbered (16), (20), (49), (50), and (51): (16)$\Rightarrow$(20), uni...

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 Aldous's self-similar random partition of the plane, determine its topological properties: in particular, do region boundaries have fractal dimens...

Does every unimodular random countable locally finite rooted graph arise as a local weak limit of finite graphs?...

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 measure-theoretic limit transition densities $p_\infty(x,y,t)$ arising from sequences of finite reversible Markov chains, construct a natural ...

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 $S_n=S_0+X_1+\cdots+X_n$ be a renewal process whose i.i.d. strictly positive recurrence times have infinite mean and a non-lattice distribution. I...

For the null-recurrent renewal process of Problem 1.1, is there a non-decreasing function $\phi$ such that $D_t/\phi(t)$ converges in distribution to ...

For the null-recurrent renewal process of Problems 1.1–1.2, assuming their answers are positive, does $(D_t/\phi(t),U_t)$ converge in distribution to ...

Let $S$ and $S'$ be random walks on $\mathbb{R}$ with the same i.i.d. step-length distribution, starting at $0$ and $x$. Suppose the step lengths are ...

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...

Let $(X,\xi)$ consist of a random element and a diffuse random measure on a locally compact second countable Abelian group. Is mass-stationarity of $(...

Does the invariant-transport characterization of mass-stationarity remain valid if the bounded jointly invariant preserving kernels are restricted to ...

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 $...

Let $(a_n)_{n\in\mathbb{Z}}$ be a locally finite configuration of points on $\mathbb{R}$. For the force law $F(x,y)=|x-y|^{-2}$, call the configuratio...

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?...

Prove that every bounded-degree transient hyperbolic graph contains a transient subtree....

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 $(G_n)$ be a bounded-degree expander family converging locally to an infinite graph $G$. Prove that the finite-graph percolation thresholds $p_c(G...

For simple random walk $(X_n)$ on the uniform infinite planar triangulation, prove that the graph distance from the starting point has exponent $1/4$,...

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 ...