The spectral matrices associated with the stochastic Darboux\n transformations of random walks on the integers

2019/07/12 by Manuel D. de la Iglesia, de la Iglesia, Manuel D., Claudia Juarez +1
Chemistry · Mathematics · Computer Science · #Molecular spectroscopy and chirality #Graph theory and applications #Matrix Theory and Algorithms

paper · pdf · doi:10.48550/arxiv.1907.05942

Abstract

We consider UL and LU stochastic factorizations of the transition probability\nmatrix of a random walk on the integers, which is a doubly infinite tridiagonal\nstochastic Jacobi matrix. We give conditions on the free parameter of both\nfactorizations in terms of certain continued fractions such that this\nstochastic factorization is always possible. By inverting the order of the\nfactors (also known as a Darboux transformation) we get new families of random\nwalks on the integers. We identify the spectral matrices associated with these\nDarboux transformations (in both cases) which are basically conjugations by a\nmatrix polynomial of degree one of a Geronimus transformation of the original\nspectral matrix. Finally, we apply our results to the random walk with constant\ntransition probabilities with or without an attractive or repulsive force.\n

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