On the average hitting times of the squares of cycles

Yoshiaki Doi; Norio Konno; Tomoki Nakamigawa; Tadashi Sakuma; Etsuo Segawa; Hidehiro Shinohara; Shunya Tamura; Yuuho Tanaka; Kosuke Toyota

arXiv · 2110.12614

On the average hitting times of the squares of cycles

Yoshiaki Doi·Norio Konno·Tomoki Nakamigawa·Tadashi Sakuma·Etsuo Segawa·Hidehiro Shinohara·Shunya Tamura·Yuuho Tanaka·Kosuke Toyota

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Abstract

The exact formula for the average hitting time (HT, as an abbreviation) of simple random walks from one vertex to any other vertex on the square $C^2_N$ of an $N$-vertex cycle graph $C_N$ was given by N. Chair [\textit{Journal of Statistical Physics}, \textbf{154} (2014) 1177-1190]. In that paper, the author gives the expression for the even $N$ case and the expression for the odd $N$ case separately. In this paper, by using an elementary method different from Chair (2014), we give a much simpler single formula for the HT's of simple random walks on $C^2_N$. Our proof is considerably short and fully combinatorial, in particular, has no-need of any spectral graph theoretical arguments. Not only the formula itself but also intermediate results through the process of our proof describe clear relations between the HT's of simple random walks on $C^2_N$ and the Fibonacci numbers.

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Yoshiaki Doi, Norio Konno, Tomoki Nakamigawa, Tadashi Sakuma, Etsuo Segawa, Hidehiro Shinohara, Shunya Tamura, Yuuho Tanaka, Kosuke Toyota. 2021-10-25. On the average hitting times of the squares of cycles. https://arxiv.org/abs/2110.12614

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