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Consider the symmetric random walk on $\{0, 1, \dots, n\}$ with transition probabilities $P(j \to j \pm 1) = 1/2$ for $1 \le j \le n-1$ and $P(0 \to 0) = P(0 \to 1) = P(n \to n) = P(n \to n-1) = 1/2$. I am interested in the spectrum of the transition matrix (which is symmetric, hence the spectrum is real).

Mathematica suggests that the characteristic polynomial is of the form $p_n(x) = (1 - x)q_n(x)$ where $q_n$ is a polynomial that is odd / even iff $n-1$ is odd / even and that has only simple zeroes. Therefore, the spectrum appears to be a symmetric set of $n$ points from the open unit interval, plus the point $\lambda = 1$..

It seems to me that this ought to be well known. In particular, the factors $q_n(x)$ in the characteristic polynomials ought to be special. However, I don't have my copy of Feller vol. I at hand, which would be the first place to look.

Does anybody know more?

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