Exponential localization conjecture for pointwise sampling functions on the Sierpinski gasket

Let $\beta_m$ be the level-$m$ vertex set of the Sierpinski gasket, let $y$ be a non-boundary vertex in $\beta_m$, and let $\phi_y^{(m)}$ be the pointwise sampling function satisfying $\phi_y^{(m)}(x)=\delta_{yx}$ for $x\in\beta_m$. For $x\in SG$, let $d_m(x,y)$ be the smallest number of level-$m$ cells separating $x$ and $y$. Exponential localization conjecture. There exist constants $c$ and $\alpha<1$, with $\alpha$ numerically about $1/3$, such that

$$\left|\phi_y^{(m)}(x)\right|\le c\alpha^{d_m(x,y)}$$

for all non-boundary vertices $y$ in $\beta_m$, all $x\in SG$, and all $m$. This conjecture, attributed in the source to earlier work on pointwise sampling functions, describes the rapid spatial decay observed numerically for both pointwise and cell sampling functions.