Renormalization of oscillator lattices with disorder

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 79, Issue 5, 2009, Pages

  Östborn, Per ^a  

^a LUND UNIVERSITY   (Sweden)

Author keywords

[No Author keywords available]

Indexed keywords

CRITICAL DIMENSION; CRITICAL PROPERTIES; LOWER BOUNDS; OSCILLATOR MODEL; REAL-SPACE RENORMALIZATION; RENORMALIZATION; SECOND-ORDER PHASE TRANSITION;

CONTROL THEORY; OSCILLATORS (ELECTRONIC);

PHASE TRANSITIONS;

EID: 67149104087     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.79.051114     Document Type: Article

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54. If we let g→, then almost all hk → and var [hk] will not exist. Those hk =g l nk flk (l, k) that might still be finite are those for which the phases in the flk s are fine tuned so that the sum vanishes.
55. Taking the ensemble variance of both sides of Eq. 21 at the fixed point, we get var [d k /dt] = (σω2) + var [hk] +2 cov [ωk, hk]. Because of the statistical inequality | cov [X,Y] | ≤ var [X] var [Y] we see that var [d k /dt] exists since (σω2) and var [hk] do so by assumption [Eq. 23]. It follows that E [d k /dt] also exists. These statements hold for any t. We may express Ωk = d k /dt t. It is straightforward to show that E [Ωk] = E [d k /dt] t and that var[Ωk] ≤max { var [d k /dt] } t Finally, we must have var [Ωk] >0 at a critical fixed point corresponding to a transition to partial synchronization.