A Markov Chain Arising from the Hopf Square Map on a Non-cocommutative Quantum Group

TL;DR

The paper constructs and analyzes a Markov chain arising from the Hopf square map on the non-commutative, non-cocommutative quantum group $U_q(\mathfrak{sl}_2)$. It identifies a graded sub-Hopf algebra with basis $\{E^{i}K^{l}\}$ and derives explicit transition probabilities $T(i,i+1)=\frac{1}{q^{i}+1}$ and $T(i,i)=\frac{q^{i}}{q^{i}+1}$, leading to a one-dimensional growth process whose distribution involves the $q$-Pochhammer symbol. A key contribution is the demonstration of a phase transition at the deformation parameter $q$, with sublinear growth for $q>1$ and linear growth for $0<q<1$, proven via martingale methods and complemented by bounds and a generalization to $X_n^{\alpha}$. These results extend the Hopf-square Markov-chain framework to quantum groups and illustrate how algebraic structure governs probabilistic behavior, with potential extensions to other quantum groups and higher-gradings.

Abstract

Expanding upon the rich history of algebraic techniques in probability, we show the existence of and construct a Markov chain using the Hopf square map on a quantum group that is both non-commutative and non-cocommutative. This extends the work of Diaconis, Pang, and Ram to other Hopf algebras. The new, one-dimensional chain requires different analytical approaches. In this case we use standard martingale theory to prove the existence of a phase transition and prove bounds on the expected growth rates.

Theorems & Definitions (21)

• Theorem 1
• Theorem 2
• Theorem 3: Diaconis, Pang, Ram
• Theorem 4
• Theorem 5
• Theorem 6
• Corollary 7
• proof
• Theorem 8
• Corollary 9