Random walk and chaos of the spectrum. Solvable model

✍ Scribed by Leonid Malozemov

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 739 KB
• Volume: 5
• Category: Article
• ISSN: 0960-0779
• DOI: 10.1016/0960-0779(94)00221-b

✦ Synopsis

We consider the spectrum of the Laplacian corresponding to the random walk on the fractal graph depending on parameter /3 > 0. The spectrum of this Laplacian is given by the iteration of the polynomial R(/l, x) = -(/l + 2)x(x -2) and the Julia set of this polynomial is the main part of the spectrum and it has a Cantorian nature. We prove that the Lebesgue measure of the spectrum is equal to zero for any /3 > 0. We consider the character of the spectrum for B + 0~ when the spectrum concentrates around two points 0, 2 and 1 is an isolated eigenvalue of infinite multiplicity. If /3+ 0 the spectrum u(-A) approaches the segment [0, 21. We prove that the spectral dimension d, is equal to 2log3/log2@+ 2) and it converges to the Hausdorff dimension of the space dH = log 3/lag 2 for fl+ 0.