Diagonal gap condition

Appendix 1. Estimates of Eigenvectors for Diagonally Dominant Matrices with Diagonal Gap Condition Appendix 2. Time Separation and Averaging in Cycles... 

APPENDIX 1. ESTIMATES OF EIGENVECTORS FOR DIAGONALLY DOMINANT MATRICES WITH DIAGONAL GAP CONDITION... 

This smallness with respect to all constants is required only in a very special case when the additional reaction has a form A, Aj (with the rate constant fc, ) and there is no reaction of the form A,. .. in the nonperturbed system. In Section 7 and Appendix 1 we demonstrated that if in a nonperturbed acyclic system there exists another reaction of the form A,. .. with rate constant k,-, then we need inequality fcy, k, only. This inequality allows us to get the uniform estimates of eigenvectors for all possible values of other rate constants (under the diagonally gap condition in the nonperturbed system). 

We have numerically diagonalized small spin systems containing up to 5 by 5 spins subjected to a small random field hf j flatly distributed in the interval (—(5/2, (5/2). We see that indeed the gap closes rather fast away from the special Jx = Jy point (Fig. 3) but remains significant near Jz = Jx point where it clearly has a much weaker size dependence. Interestingly, the gap between the lowest 2ra states and the rest of the spectrum expected in the limits Jz 33> Jx or Jz -C Jx appears only at Jx/ Jz > jc with a practically size independent jc 1.2. We also see that the condition l 1 eliminates all low lying states in the Jz lowest excited state in l 1 sector... 

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