Mathematics Problem Archive

For a $d\times d$ real Gaussian matrix $G_{\mathbb{R}}$ and complex Gaussian matrix $G_{\mathbb{C}}$, both normalized to entry variance $1/d$, define ...

Let ${W}$ denote a symmetric Wigner matrix with i.i.d. entries ${W_{ij}\sim \mathcal{N}(0,1)}$ . Also, given ${B\in\mathbb{R}^{n\times n}}$ symmetric,...

Prove or disprove: there exists a universal constant $C$ such that, for any choice of $n$ symmetric matrices $H_1,\dots,H_n\in\mathbb{R}^{n\times n}$ ...

Let $X_t$ be two-dimensional Brownian motion. (i) For every pair $x,y \notin X[0,1]$, is there a Jordan arc $\Gamma$ containing $x$ and $y$ such that ...

For a set $B$, define its percolation dimension as the infimum of the Hausdorff dimensions of Jordan arcs $A\subset B$ containing at least two distinc...

Let $D$ be a triangle whose angles are all strictly less than $\pi/2$, and let $\mu_2>0$ be the second eigenvalue of the Laplacian on $D$ with Neumann...

Let $D\subset\mathbb{R}^2$ be a connected open set with smooth boundary, and let $X,Y$ be synchronously coupled reflected Brownian motions in $D$ driv...

For each $k\in\mathbb{Z}$, let $B^k$ be Brownian motion and $T_k$ a stopping time, with the stopped pieces independent, $0\le T_k<\infty$, and with th...

Choose uniformly a bijective labeling of the vertices of the $n\times n$ discrete square by $1,2,\ldots,n^2$, and call a vertex a peak when all adjace...

On $\mathbb{Z}^2$, take nearest-neighbor jump probabilities $p_1,q_1,p_2,q_2$ in directions $\pm e_1,\pm e_2$, with $p_1>q_1$ and $p_2>q_2$. If the an...

Give a mathematically useful definition of a process being a 'martingale for practical purposes', so that failure means it is practical to find a stop...

Find a simple network-and-demand toy model in which the marginal satisfiability proportion $r(t)$ can be calculated analytically and exhibits the prop...

For sparse $n$-vertex graphs of average degree $O(1)$ whose vertices have distinct $O(\log n)$-length labels over a finite alphabet, construct univers...

Obtain relaxation- and mixing-time bounds for reversible coagulation-fragmentation Markov chains on finite sets in terms of their model parameters....

For the two stationary branching-coalescing models on $\mathbb{Z}^3$ described by Aldous, prove that as particle intensity tends to zero the suitably ...

Study the constrained Ising storage model described on the page when the underlying graph itself changes in time....

Tasks arrive as a rate-one Poisson process and have types in $[0,1]$; batch processing time $S$ is monotone and strictly subadditive and type $a$ incu...

For the Metropolis chain on a finite Cayley graph with stationary law $\mu(p)$ obtained by stopping random walk at a geometric time, analyze its relax...

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?...

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 $\alpha>1$...

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...

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...

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 ...