17 problems

Optimal Power of Few conjecture. There is a function $f(x)$ such that $f(x)>1/2$ for $x>0$, $f(x)$ tends to $1$ as $x$ tends to infinity, and, in majority dynamics on $G(n,p)$ with…

Let $G$ be an infinite graph of bounded degree, and choose the initial opinions independently and uniformly from $\{-1,1\}$. Under majority dynamics, each node's opinion almost sur…

Majority Dynamics conjecture. For every $\varepsilon>0$, with high probability the number of oscillating vertices lies in

Let $\mathbb{G}(2n+\Delta,1/2)$ be the random graph with vertex sets initially colored red and blue as … A color wins when all vertices eventually have that color. Tran–Vu's conjec…

Benjamini–Chan–O'Donnell–Tamuz–Tan conjecture. With probability $1-\varepsilon$, the vertices in $G(n,p)$ have an $(1-\varepsilon)$-proportion agreement after sufficiently many day…

Let $G(n,p)$ be a random graph, let $P_t$ and $N_t$ denote the sets of vertices playing the two strategies at time $t$, and let $d$ be the parameter in the condition $p\geq d/n$. C…

Let $G_{n,p}$ be a random graph sampled independently of an arbitrary initial red-blue coloring of its vertices, and let $\varepsilon>0$. Assume … After the first step of majority…

Let $G_{n,p}$ be a random graph on $n$ vertices, and consider majority dynamics from an initial red-blue coloring with more red vertices than blue. For fixed $p\in(0,1)$, let…

Let $c1(k,n)$ be the probability that Red wins when Red's camp initially has size $n/2+k$, with $p=.5$, and define the value of the $i$th defector by … with $c1(n,0)=1/2$. Values o…

Let $\rho(k,n)$ be the probability that Red wins when Red's camp initially has size $n/2+k$, with edge probability $p=.5$. The power of one conjecture. There is a constant $b4>0$ s…

Let $\operatorname{MD}_{\mathrm{coins}}$ denote the coin-valued median dynamics process on $\mathbb{Z}^d$, and let $\eta_t(\underline{0})$ be the opinion at the origin. Coin-median…

Let $\operatorname{MD}_{\mathrm{coins}}$ denote the coin-valued median dynamics process on $\mathbb{Z}^d$, and let $\eta_t(\underline{0})$ be the opinion at the origin. Coin-median…

Let $(\eta_t)_{t\geq0}$ be median dynamics and, for an initial density $p\in[0,1]$, let $(\xi_t^p)_{t\geq0}$ be the coupled majority-dynamics process. Say that majority dynamics fi…

Let $p_c(G)$ be the infimum of the densities $p\in[0,1]$ for which zero-temperature Glauber dynamics on $G$, started from i.i.d. $\operatorname{Bernoulli}(p)$ opinions, converges a…

Let $G(n,p)$ be a random graph with vertex set $V_n$, let $d=np$, and let $S_t(v)\in\{-1,+1\}$ be the state of vertex $v$ after $t$ rounds of synchronous majority dynamics. Benjami…

For majority dynamics on $\mathbb{Z}^2$, let $\eta_\infty$ be the pointwise limiting configuration and define … Let $p_c(t)$ be the critical percolation probability at time $t$. Li…

Information retention conjecture. For every such sequence and every $\frac12<p<1$,