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: $\int_{T^2} V^ν_\varepsilon \, dν\le C_τ\varepsilon^{1-τ} {\mathcal E}[ν]^τ |ν|^{1-τ}$ is valid on bi-tree with any $τ>0$. We show that unlike the estimate on a simple tree, one cannot make $τ=0$ on bi-tree. On tri-tree we know that the previous estimate (the surrogate maximum principle) is valid with $τ=2/3$. We do not know any such estimate with any $τ<1$ on four-tree. The third counterexample disproves the estimate $\int_{T^2} V^ν_x \, dν\le F(x)$ for any function $F$ whatsoever for some probabilistic $ν$ on bi-tree $T^2$. On a simple tree $F(x)=x$ would always suffice to make this inequality to hold.