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The second largest eigenvalue of a transition matrix $P$ has connections with many properties of the underlying Markov chain, and especially its convergence rate towards the stationary distribution.
Symmetric random walks on groups
Harry Kesten · 1959
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F.L. Bauer and C.T. Fike · 1960
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The eigenvalues of random symmetric matrices
Zoltán Füredi and János Komlós · 1981
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Random graphs
Béla Bollobás · 1985
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Noga Alon · 1986
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Eigenvalue bounds on convergence to stationarity for nonreversible markov chains, with an application to the exclusion process
James Allen Fill · 1991
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Pierre de la Harpe, A. Guyan Robertson, and Alain Valette · 1993
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Persi Diaconis · 1996
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Random graphs with arbitrary degree distributions and their applications
M. E. J. Newman, S. H. Strogatz, and D. J. Watts · 2001
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The size of the largest strongly connected component of a random digraph with a given degree sequence
C Cooper and A Frieze · 2004
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Non-backtracking spectrum of random graphs: community detection and non-regular Ramanujan graphs
C. Bordenave, M. Lelarge, and L. Massoulié · 2015
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A new proof of Friedman’s second eigenvalue Theorem and its extension to random lifts
C. Bordenave · 2015
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The circular law for signed random regular digraphs
N. A. Cook · 2015
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Spectrum of large random Markov chains: heavy-tailed weights on the oriented complete graph
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Eyal Lubetzky and Yuval Peres · 2016
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