14 problems

Stabilizability critical-density conjecture. If $\rho<\rho_c$, then $\nu$ is stabilizable; if $\rho_c<\rho\leq 2d$, there exist stationary and ergodic measures $\nu$ that are not s…

Let $G=(V\cup\{s\},E)$ be a finite graph with designated sink vertex $s$, and let $\rho_G(p)$ denote the stationary average density for the $p$-toppling rule, where $p\in[0,1]$ and…

Let $\zeta_{\mathtt{DD}}$ be the limiting density of the driven-dissipative abelian sandpile model on increasingly large finite boxes, and let $\zeta_{\mathtt{FE}}$ be the critical…

Equal-signature conjecture. The numbers of CNATs whose underlying permutation has even and odd determinant (signature) are equal.

Maximum periodicity conjecture. The maximum periodicity of any vertex $v$ in each listed fractal graph is given by the corresponding value above.

Biggs's conjecture. The Biggs–Merino polynomial is equal to

Perrot–Pham's conjecture. The polynomial $\mathcal B(G,s;y)$ is independent of the choice of the sink vertex $s$.

A cactus graph here is a graph obtained by decorating a regular tree with connected transitive graphs, and the cell-wise first-wave critical exponent measures the power-law decay o…

Let $\Gamma'$ be a decoration of a $d$-regular tree, and let $g(x)$ be the generating function for weighted clusters containing the origin. Let $D$ denote the exponential growth fa…

Let $\Gamma$ be an infinite $d$-regular tree, and let $\Gamma'$ be a graph quasi-isometric to $\Gamma$, including a decoration of $\Gamma$. The cell-wise first-wave critical expone…

Let $\Gamma$ be an infinite $d$-regular tree, and let $\Gamma'$ be a graph quasi-isometric to $\Gamma$, including a decoration of $\Gamma$. A critical exponent is the power-law exp…

A critical exponent describes the power-law decay of avalanche-mass probabilities in an increasing graph sequence exhausting an infinite graph. Let $\Gamma$ be an infinite graph an…

Avalanche mass exponent conjecture. The critical exponent for the mass of an avalanche is

Logarithmic two-point correlation conjecture. Given the conjectured field identifications, for every $h\geq 2$ the dominant term is