29 problems

Let the frog model start with one frog per vertex on the homogeneous tree $\mathbb{T}_d$ of degree $(d+1)$, with frogs performing the model's random walks and waking sleeping frogs…

Fix a maximum degree $d$. Let $(G_n)_{n\geq1}$ be a sequence of finite connected graphs, with $G_n=(V_n,E_n)$, $|V_n|\to\infty$, and maximum degree at most $d$. The polylogarithmic…

Let $G$ be a vertex-transitive graph with superlinear growth, and let $t_{\mathrm{c}}(\lambda)$ denote the critical lifespan for particle density $\lambda$. Positive critical lifes…

Let $G$ be a fixed graph, and let $p_{\mathrm{c}}(G)$ denote its Bernoulli percolation critical parameter. Assume $p_{\mathrm{c}}(G)=1$, and let $\lambda_{\mathrm{c}}(t)$ and…

Let $G$ be a non-amenable unimodular vertex-transitive graph, and consider the infinite-lifespan frog model. Write $\lambda_{\mathrm{c}}(\mathrm{recurrence})$ for the critical dens…

Let $G$ be an amenable transitive graph, let $\lambda>0$, and let $\mathcal{C}_{\mathbf{0}}$ denote the set of vertices reached from the origin in the infinite-lifespan frog model.…

Let $G$ be a transitive graph of superlinear growth, and let $\lambda_{\mathrm{c}}(t)$ and $t_{\mathrm{c}}(\lambda)$ denote the critical curves of the frog model. Strict monotonici…

Let $G$ be a transitive graph, and let the frog model on $G$ have critical curves $\lambda_{\mathrm{c}}(t)$ and $t_{\mathrm{c}}(\lambda)$. Continuity conjecture. The phase transiti…

Let $G=(V,E)$ be a transitive graph with superlinear growth. For $x uin V$, let $Bx(n)$ denote the ball of radius $n$ centered at $x$, and write $gx(n)=|Bx(n)|$. The graph has **su…

Let $\operatorname{FM}(d,p,\nu)$ be the frog model on the rooted $d$-ary tree with drift parameter $p$ and initial particle-distribution parameter $\nu$. Let…

Hoffman–Johnson–Junge conjecture. The frog model $\operatorname{FM}(3,1/4)$ is recurrent.

Beckman–Hoffman–Johnson conjecture. The critical drift satisfies

Monotonicity conjecture. For all $d\ge 2$,

Transience conjecture. There exists a constant $c=c(\mathsf{GW})>0$ such that $\mathrm{FM}(\mathsf{T},\eta,\mathsf{SRW})$ is transient for $\mathsf{GW}$-almost all infinite realiza…

Strict shape inclusion conjecture. If $p_1<p_2$, then

Shape-determined coexistence conjecture. For bounded initial sets, the possibility of coexistence is determined by the relation between the one-type shapes … .

Let $\mathbb Z^d$ be the integer lattice and let $\vec{\mathbb Z}^d$ be its oriented version. The critical frog model $\operatorname{CFM}(G,m)$ places $m$ sleeping frogs at each ve…

The critical Brownian frog model is obtained by placing particles according to a Poisson point process in $\mathbb R^d$, attaching overlapping disks of radius $r$ to the particles,…

Continuous-time coexistence conjecture. For $\eta(x)\equiv 1$, coexistence is possible if and only if

Shape-coincidence conjecture. Coexistence is possible if and only if the one-type shapes coincide:

Let $d\geq2$ and $\lambda>0$. Let $\mathfrak{G}(n,d)$ be the collection of all connected $n$-vertex $d$-regular graphs, and let $H_{d,n}$ be the graph defined in the source by arra…

The effective-resistance susceptibility conjecture. There exists a non-decreasing, diverging function $f:\mathbb{R}_+\to\mathbb{R}_+$ such that

Let $G=(V,E)$) be a finite connected vertex-transitive graph. Write $P$ for the transition matrix of simple random walk on $G$, let $ut:=minvsum{i=0}^{t}P^i(v,v)$, and define … Le…

Let $G$ be a connected non-amenable graph of bounded degree, and let $\lambda$ be the frog-model particle density. Critical-density recurrence conjecture. There exists…

Let $G=(V,E)$ be a finite connected vertex-transitive graph, and let ${\mathcal S}(G)$ denote its frog-model susceptibility. Vertex-transitive concentration conjecture. For every s…