Mathematics Problem Archive

Let $X$ be a centered Gaussian random vector in $\mathbb{R}^n$ with known covariance matrix, and for an orthonormal basis $B=(b_1,\ldots,b_n)$ let $N_...

Part (3) of the problem: Let $s\leq d\leq m$ and $z_1,\dots,z_m\in \mathbb{R}^d$ i.i.d. random vectors with i.i.d. entries $\left( z_k\right)_j = \lef...

Give a tight upperbound to $\mathbb{E}\left\| A^{\otimes k} -\mathbb{E} A^{\otimes k} \right\|.$...

Is the k-medians Linear Programming relaxation tight even for point clouds coming from generative models that do not have a community structure?...

Can one give conditions for integrality of the k-medians LP or the k-means SDP based on stability type properties (on the fact that the data is “well-...

Is the Semidefinite programming relaxation for the positive Principal Component Analysis problem tight with high probability for Wigner matrices?...

Let $X_t$ be $d$-dimensional Brownian motion starting at the origin, let $D$ be an open subset of $\mathbb{R}^d$ containing the origin, and let $\tau=...

Let $N$ particles move as independent Brownian motions in a bounded connected open set $D\subset\mathbb{R}^d$. Whenever a particle hits the complement...

Let $D\subset\mathbb{R}^d$, $d\ge2$, be bounded, connected, and open. Suppose there are coupled reflected Brownian motions $X_t,Y_t$ in $D$ and $\vare...

For the exclusion process on $\mathbb{Z}$ with $p(x,y)=p(y-x)$, assume $\sum_x|x|p(x)<\infty$, $\sum_xxp(x)>0$, and $\sum_{x<0}x^2p(x)=\infty$. Does t...

For the exclusion process on $\mathbb{Z}^d$ with translation-invariant kernel $p(x,y)=p(y-x)$ and zero mean $\sum_xxp(x)=0$, prove that every stationa...

For nearest-neighbor asymmetric exclusion on $\mathbb{Z}$ with $p(1)=p>q=p(-1)$, start from the deterministic configuration $\cdots11110000\cdots$. Is...

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...

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...

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

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

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 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 $(X,\xi)$ 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 ...

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...

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....

In one-dimensional long-range percolation with edge probabilities proportional to $\beta|i-j|^{-2}$, study the distance exponent $\theta(\beta)$ defin...

Let $D_n$ be the source-to-sink first-passage distance in the recursively substituted hierarchical graph whose distances satisfy $D_n\stackrel d=D_{n-...

For the hierarchical first-passage distances $D_n$ satisfying $D_n\stackrel d=D_{n-1}+\min(D'_{n-1},D''_{n-1})$, determine concentration around the me...

For i.i.d. first-passage percolation on $\mathbb{Z}^2$, prove or disprove that boundary fluctuations have a Tracy–Widom limit and that the variance of...

Prove that natural i.i.d. first-passage-percolation models on $\mathbb{Z}^d$, including exponential edge lengths, almost surely contain no two-sided i...

Determine the almost-sure asymptotic growth rate of the effective resistance from the root to graph-distance $r$ in the uniform infinite planar triang...

Let $G$ be a distributional local limit of finite planar graphs. Prove that $p_c^{\mathrm{site}}(G)\ge1/2$ almost surely and that there is no infinite...

On the $n\times n$ grid with a Gaussian free field with no boundary conditions, give every vertex length equal to the exponential of the field. If $\g...

For every $p\ge4$, determine the joint law of the matrix of pairwise distances among $p$ independent points sampled from the volume measure of the Bro...

Let $G$ be a connected quasi-transitive graph and let $p\in(0,1)$. If Bernoulli percolation has more than one infinite cluster almost surely, prove th...

Give general conditions implying $p_u(G)<1$. In particular, prove or disprove that every one-ended transitive graph has $p_u(G)<1$....

For nearest-neighbor independent bond percolation on $\mathbb{Z}^d$, $d\ge2$, let $p_c(d)$ be the critical edge-retention probability. Prove that at $...