Mathematics Problem Archive

For the posterior random weighted graphs $G(t)$ defined from independent Poisson edge counts and flat priors, study the process $\operatorname{gap}(G(...

For sequences of finite weighted networks with vertex recovery rates and stationary SIS infection counts $X^{(n)}_{\theta,\varepsilon}$ satisfying the...

For random proximity graphs on a planar Poisson point process, determine rigorous orders of magnitude for the transversal deviation $T_r$ of a shortes...

For both the diagonal-flip chain on triangulations of the regular $n$-gon and the branch-rotation chain on $n$-cladograms, prove that the relaxation t...

On the bidirected torus $\mathbb{Z}_N^d$ with fixed $d\ge3$, let $b^{(N)},t^{(N)},m^{(N)}$ count excursions of a uniform Eulerian circuit longer than ...

For a uniform Eulerian circuit on the bidirected two-dimensional torus, does $\log L_2^{(N)}/\log N$ converge in distribution to a random variable wit...

On the bidirected complete $n$-vertex graph, is the expected number of length-$i$ excursions in a uniform Eulerian circuit asymptotic to $e^{-i/n}$?...

For a uniform Eulerian circuit on the bidirected Hamming cube $\{0,1\}^d$, determine the asymptotic behavior or distribution of the shortest excursion...

Is there a relation between space-filling $\operatorname{SLE}_\kappa$ for $\kappa>8$ and the conjectural continuum limit of uniform Eulerian circuits ...

Improve the explicit upper and lower bounds for the minimum network length functions $\Psi^{ave}(s)$ and $\Psi^{worst}(s)$, and prove whether there is...

For two independent random $n$-cladograms, under both the uniform and coalescent distributions, prove $\mathbb{E}C_n=n^{\gamma+o(1)}$ for respective c...

For two independent coordinatewise partial orders generated by uniform points in the unit square, prove $\mathbb{E}C_n\sim c n^{1/3}$ and establish th...

For continuous-time processes that merge adjacent polygonal planar regions $A,B$ at a geometry-dependent rate $r(A,B)$, give sufficient conditions on ...

When every adjacent pair of planar empires merges at rate $r(A,B)=1$, does percolation occur?...

For the city-growth model, prove: (a) if $\alpha>1$, the eventual number of cities $M(\infty)$ is finite almost surely; (b) if $\beta<2\alpha$, the la...

In the balanced regime $0<\alpha<1$ and $\beta>2\alpha$, prove that the upper and lower growth exponents for influence and city population all equal $...

If $\alpha=1$ and $\beta>2$, prove $N_{(1)}(t)=t(\log t)^{1-2/\beta+o(1)}$ almost surely....

For the associated influence-cell dynamical system in general position with positive initial weights, prove that one weight tends to $1$ if $\alpha>1$...

Construct a scale-invariant random spatial network whose law is mathematically natural, for example with an explicit formula for the distribution of $...

Construct a scale-invariant random spatial network that is visually realistic, in the sense of not looking very different from a real-world road netwo...

Determine the set of possible triples $(\Delta=\mathbb{E}D_1,\ell,p(1))$ over all scale-invariant random spatial networks....

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...

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 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 $X$ 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 $...

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 $k$-...