This talk will detail various collaborative projects I have been working on related to explicit realizations of N soliton and g gap solutions to the Korteweg—de Vries equation, their relevance in the statistical study of soliton gasses, and the use of finite gap operators to derive of mixing time bounds for Markov chains generated by a periodic Jacobi—Markov operator (i.e. a tridiagonal Markov operator). This talk will begin with a literature review of recent experimental results on solitons. I will then discuss multi-soliton and periodic solutions to the KdV equation The last topic covered will be the calculation of mixing time bounds for periodic birth and death Markov chains generated by periodic Jacobi—Markov operators. I will end with an example of the calculation of mixing time bounds using an elliptic Baker—Akheizer function for Jacobi—Markov operators with period two, but the method generalizes by the use of hyper-elliptic Baker—Akheizer functions for periodic Jacobi—Markov operators.