5 problems

Let $m\geq 2$, let $\rho\in[0,1]$, and let $\varepsilon>0$. For a function $f:\{1,\ldots,m\}^n\to\Delta_m$, write $\operatorname{Inf}_i(f_j)$ for the influence of coordinate $i$ on…

Theorem 5.1 conjecture. Theorem 5.1 is true in all $d\ge2$.

Let $0\leq\rho<1$ and $\epsilon>0$. Let $f:\{-1,1\}^n\to\{-1,1\}$ satisfy ${\bf E}[f]=0$, and write $\mathrm{Inf}_i(f)$ for the influence of coordinate $i$. Let $V\subseteq[n]$ inc…

In one spatial dimension, let $\Phi$ be the global solution with $a=b=0$, and regard it modulo additive constants so that it has stationary spatial increments. Brownian limit conje…

Brownian-motion conjecture. Under assumptions … , Theorem $$ can be strengthened so that the random walk on a discrete point process, under appropriate scaling, converges to Browni…