Differences between the potential theories on a tree and on a bi-tree

Abstract

In this note we give several counterexamples. One shows that small energy majorization on bi-tree fails. The second counterexample shows that partial energy estimate always valid on a usual tree by a trivial reason (and with constant $C=1$) cannot be valid in general on bi-tree with any $C$ whatsoever. On the other hand, a weaker partial energy estimate called surrogate maximum principle: $\int_{T^2} V^\nu_\varepsilon \, d\nu \le C_\tau \varepsilon^{1-\tau} {\mathcal E}[\nu]^{\tau} |\nu|^{1-\tau}$ is valid on bi-tree with any $\tau>0$. We show that unlike the estimate on a simple tree, one cannot make $\tau=0$ on bi-tree. On tri-tree we know that the previous estimate (the surrogate maximum principle) is valid with $\tau=2/3$. We do not know any such estimate with any $\tau<1$ on four-tree. The third counterexample disproves the estimate $\int_{T^2} V^\nu_x \, d\nu \le F(x)$ for any function $F$ whatsoever for some probabilistic $\nu$ on bi-tree $T^2$. On a simple tree $F(x)=x$ would always suffice to make this inequality to hold.