Mathematics Problem Archive

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 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 ...

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...

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 $(G_n)$ 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 ...

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....