11 problems

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 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…

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…

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…

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

For $p\in[0,1]$, let the frog model with drift parameter $p$ have maximum speed, namely the limiting speed of its rightmost frog, denoted by $v(p)$. Concavity conjecture. The speed…

Recurrence–transience conjecture. The frog model on $\mathbb{T}_d$ is recurrent almost surely for $d=3$ and transient almost surely for $d=4$.