29 problems

Two-dimensional Weil conjecture. For every endomorphism $M\in\operatorname{End}_{\mathcal C_{\mathbb F_q}}(X)$ there exist super Boltzmann data $(V_1,V_2,R)$ such that, for every f…

Let $G$ be the infinite nearest-neighbor graph on the integer lattice in $d$ dimensions, and let ${\bf T}$ be the uniform spanning forest obtained as a distributional limit of unif…

Let $f(w)$ be the spanning-forest free energy per site on a regular lattice of dimension $d\geq2$ and coordination number $r$, and let $f_0$ denote the entropy per site for spannin…

Critical-fugacity asymptotic conjecture.

Let $J_{\mathrm{rect}}(x)$ denote the maximum number of points achievable for a binary word $x$ in the two-dimensional rectangular lattice. For binary words $x$ and $y$, Stecher's…

Let $\mathcal{C}_p$ be the class of probability distributions on $\Omega$ that are independent and identically distributed over the vertices, isotropic, and have expected degree eq…

Let $\mathsf{B}^{n+1}$ be a background phase and let $\mathsf{C}^{n+1}$ be an $(n+1)$D anomaly-free topological order. An MCLP lattice model over $\mathsf{B}^{n+1}$ has a local spi…

For $d\geq3$, integers $n\geq2$ and $m\geq3$, define … where $\|x_1-x_2\|$ is graph distance. Separation-distance conjecture. (1) There exist $C(d,m),c(d,m)>0$ such that, for all…

For $d\geq3$ and $n\geq2$, let $\mathfs{M}_n(\mathbb{Z}^d)$ be the least positive harmonic measure among subsets of $\mathbb{Z}^d$ of cardinality $n$. Two lattice vertices are $*$-…

Let $\psi(\mathbb{Z}^2)$ denote the reciprocal extremal factor governing the exponential lower bound for positive harmonic measures on finite subsets of $\mathbb{Z}^2$. The propose…

Let $\theta^{\rm D}(k,d)$ be the probability of an infinite directed path starting at the origin in the directed $k$-neighbor graph on $\mathbb{Z}^d$, and let…

For $d\in\mathbb{N}$, let $\theta^{\rm X}(2,d)$ denote the percolation probability of the $2$-X-neighbor graph on $\mathbb{Z}^d$. Two-neighbor X-graph conjecture. The $2$-XnG perco…

Let $\theta^{\rm X}(k,d)$ denote the percolation probability of the $k$-X-neighbor graph on $\mathbb{Z}^d$. X-neighbor monotonicity conjecture. For fixed $d$, the map…

Let the $k$-BnG be the bidirectional $k$-neighbor graph on $\mathbb{Z}^d$, and let $\theta^{\rm B}(k,d)$ denote its percolation probability. Bidirectional threshold conjecture. The…

Let $\theta^{\rm D}(k,d)$ denote the percolation probability of the directed $k$-neighbor graph on $\mathbb{Z}^d$. Fixed-k dimension monotonicity conjecture. For all $k,d\geq 2$ wi…

For each $d\in\mathbb{N}$, let the $2$-DnG be the directed $2$-neighbor graph on $\mathbb{Z}^d$, and let $\theta^{\rm D}(2,d)$ be its percolation probability. Two-neighbor directed…

Let $\theta^{\rm D}(k,d)$ denote the probability that the origin belongs to an infinite directed cluster in the directed $k$-neighbor graph on $\mathbb{Z}^d$. Directed 2-neighbor p…

Let $\theta^{\rm D}(k,d)$ denote the probability that the origin belongs to an infinite directed cluster in the directed $k$-neighbor graph on $\mathbb{Z}^d$. Dimension monotonicit…

For each $d$, let $t_{n,d}$ denote the number of tiles contained in $[n]^d$, and let $v_n$ denote the volume parameter used in the paper. Polynomial-factor conjecture. There is a c…

Fix an integer $k$, and pick $x \in [n]^d$ uniformly at random. Let $\mathcal{S}$ be a random tile contained in $[n]$, and let $B_k$ be the $k$-ball around $x$. Local Bernoulli lim…

The critical-exponent asymptotic conjecture. For every $d\ge2$, such a critical exponent exists, and

Exponential lower-bound conjecture. For every dimension $d\geq 3$,

Square-spiral conjecture. Asymptotically, the square spiral realizes the least positive value of harmonic measure, in the sense that

Let $\varphi$ be an anisotropy and let $\alpha$ be a suitable parameter. For the discrete solutions $\{E^k\}$ of scheme … . Conjecture on crystalline growth. Under suitable assumpt…

Independent-model phase-diagram conjecture. There exists $\lambda_{\mathrm{c}}(p,d)\in(0,1)$ such that $\lambda_{\mathrm{c}}(\mathord{\cdot},d)$ is continuous and strictly increasi…