44 problems

Tail asymptotics conjecture. For $k$ sufficiently large,

Let $M$ be a response matrix, and let $\det M_A^B$ be a circular minor, meaning that at least one of the index sets $A$ and $B$ is contiguous. Such a minor is called a semicontiguo…

Let $G_k(n)$ be the $k\times n$ grid graph, with vertices denoted $[i,j]$, and let $R(k,n)$ be the electrical resistance between $[1,1]$ and $[k,n]$ when every edge has resistance…

Let $G$ be a graph with nonnegative edge conductances ${\bf x}=\{x_e\}_{e\in E}$, and let $i,j$ be distinct vertices. Write ${\mathcal Y}_{ij}({\bf x})$ for the effective conductan…

Let $G$ be a finite transitive graph, and let the effective electric resistance between two vertices be computed in the graph's electrical network. Bounded-resistance conjecture. T…

Let $G_{m,n}$ be the degree-$m$ part of the grove algebra, and let $c$ generate the cyclic action … The basis elements are to be indexed by height $m$ staircase plane partitions of…

Kenyon–Wilson's conjecture. The matrix $M$ is the response matrix of a circular electrical network belonging to a cell of co-dimension $r$ in the poset $EP_n$ if and only if the fi…

Let $\mathcal{E}$ be a well-connected network with an odd number of boundary nodes and response matrix $M_R(\mathcal{E})$. Let $A(\mathcal{E})$ be the matrix constructed from this…

Limiting average-resistance conjecture. The sequence $M_n$ converges to $1.25$ as $n$ grows.

Let $\mathcal{X}_n$ be the space of electrical networks, let $\mathcal{X}_\tau$ be the electroid variety indexed by a matching $\tau$ on $2n$ points, and let…

Let $G_{d,n}$ be the degree-$d$ homogeneous piece of the electrical network coordinate ring, let $\mathcal{C}^{(d)}_n$ index the proposed basis, let $E_{\mathbf{P}}$ denote its ele…

Let $L_s^{(i)}$, for $i\in\{1,2\}$, be the edge-resistance quantities defined in the paper, and let $\operatorname{num}(x)$ denote the numerator of the maximally reduced fractional…

Let $T_{n,k}$ denote the triangular grid obtained after $k$ transformations from the $n$-grid, and let $T_{n,n-s}^s$ denote an $s$-subgrid of $T_{n,n-s}$. For an edge, its edge val…

Edge-label ratio conjecture. For the $T_1(k,c)$ grid,

A weighted $k$-nested network is a weighted network in which the relevant nested structure has depth $k$, and its resistance distance is the pairwise effective resistance between v…

Let $({\mathbb{Z}}^d,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\,p)$ be a biased disordered random network, with $d\geq 3$ and $0<\lambda_1\leq 1<\lambda_2$. A network is…

Let $p_c$ denote the critical probability for Bernoulli bond percolation on ${\mathbb{Z}}^d$, and let $({\mathbb{Z}}^d,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\,p)$ den…

Conditional local-move existence conjecture. For every integer $n\geq 3$, there exists a conditional local move on $\Pi_n$ such that:

Hexagon conditional local-move conjecture. There is a hexagon conditional local move of the displayed form, with the conjectural conductance formulas described in the paper, that p…

Local-move completeness conjecture. Any two electrically equivalent graphs associated to rnpds are connected by the local moves stated in this section.

Let $N$ be the network from the hexwheel example, let $m\geq 3$, and write $\operatorname{cp}_m(\lambda)=\operatorname{rcp}(M(N))$. Let $\omega_\pm=\lambda-2\pm\sqrt{\lambda^2+4}$.…

Resistance exponent conjecture. The exponent $\mathfrak{r}$ exists and satisfies

Let $\g$ be a graph, let $u$ be a choice of vectors defining the associated real arrangement, and let $\eng\in(0,\infty)^k$ be a rational point. When the arrangement is the graphic…

A response matrix is the matrix describing the boundary behavior of a network. A circular planar network is a network embedded in a disk with all boundary vertices on the boundary…

Let $L$ be the response matrix of a minimal circular planar network with $m$ edges, and call a minor of $L$ noninterlaced when its row and column index sets are noninterlaced. Base…