A generalization of carries process and riffle shuffles

Abstract

As a continuation to our previous work [7], we consider a generalization of carries process. Our results are : (i) right eigenvectors of the transition probability matrix, (ii) correlation of carries between different steps, and (iii) generalized riffle shuffle whose corresponding descent process has the same distribution as that of the generalized carries process.

Determinantal Formula for Generalized Riffle Shuffle

“…Then formulas in Theorems 1.1, 1.3 follows from Lindeström-Gessel-Viennot lemma [8,6]. This bijection is the same as that between a generalized carries process called (b, n, p)-carries process, and the descent process of the random walk generated by (b, n, p)-shuffle, discussed in [12]. The contents of later sections are outlined as follows.…”

“…Diaconis-Fulman [3,4] showed that the descent process of this random walk is a Markov chain which has the same distribution as that of the carries process in adding numbers studied by Holte [7]. The authors in this paper previously considered a generalized riffle shuffle on the colored permutation group G p,n , and a generalized carries process and showed that the descent process of the former has the same distribution as the latter [11,12]. In this paper, we study the generalized riffle shuffle studied in [12] and derive a formula for the probability of finding descents at given positions.…”

“…Then [12] f is a bijection on D(b) and (i) x < y if and only if f (x) ≺ f (y), and (ii) x > c if and only if f (x) ≡ 0 (mod p). Therefore f is a bijection between…”

“…The authors in this paper previously considered a generalized riffle shuffle on the colored permutation group G p,n , and a generalized carries process and showed that the descent process of the former has the same distribution as the latter [11,12]. In this paper, we study the generalized riffle shuffle studied in [12] and derive a formula for the probability of finding descents at given positions. To describe our results, we introduce some notations and definitions.…”

“…In section 2, we prove theorems 1.1, 1.3. In Appendix, we introduce (b, n, p)-carries process and discuss some of its properties to supplement our previous works [11,12] : (i) a simplified proof for the diagonalization of the transition probability matrix, (ii) the relation between the eigenspace of the transition probability matrix for the random walk mentioned above and the right eigenvectors of that of (b, n, p)-carries process, and, (iii) we study the convergence speed of this random walk and discuss the cut off phenomenon.…”