Spectral and dynamical properties of certain random Jacobi matrices with growing parameters

Abstract

Abstract. In this paper, a family of random Jacobi matrices with off-diagonal terms that exhibit power-law growth is studied. Since the growth of the randomness is slower than that of these terms, it is possible to use methods applied in the study of Schrödinger operators with random decaying potentials. A particular result of the analysis is the existence of operators with arbitrarily fast transport whose spectral measure is zero dimensional. The results are applied to the infinite Dumitriu-Edelman model (2002), and its spectral properties are analyzed.

Fluctuations of Interlacing Sequences

“…The discussion above raises the expectation that a version of (15) with the critical pointsλ * j in place of the submatrix eigenvaluesλ * * j may bear more similarity to Kerov's theorem (12). Indeed, appealing to Johansson's central limit theorem (14), we show that …”

“…This raises the question what is the analogue of Kerov's central limit theorem (12) in random matrix context.…”

“…The right-hand side of (15) does not look similar to (12), for two reasons. First, a typical random partition of n has √ n rows, therefore the normalisation 1/ √ n in (15) would correspond to 1/ 4 √ n, rather than 1/ √ n, in (12).…”

“…First, a typical random partition of n has √ n rows, therefore the normalisation 1/ √ n in (15) would correspond to 1/ 4 √ n, rather than 1/ √ n, in (12). Second, the Gaussian process in (16) has a continuous modification (it is roughly a reparametrised Brownian motion), whereas the sum in (12) does not even converge in L 2 .…”

“…In Section 4, we return to random partitions and to the theorems of Kerov and Ivanov-Olshanski, stated above as (12) and (13). We describe the setting, and use Proposition 3.1.4 to derive one from the other.…”