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Trace for the Loewner equation with singular forcing

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L P Kadanoff and M Kleine Berkenbusch



CORRIGENDUM

This is a Corrigendum for the article 2004 Nonlinearity 17 R41

In the above invited article there was an error in the description of equation (17) which gives the behaviour of the trace close to the singularity in the case of square-root forcing β = 1/2. The equation is correct only in the limit as κ goes to infinity. The correct equation, valid for all κ > 4, is
        \gamma(s) - y_- = C \rme^{\rmi\phi} [\xi(s)]^{({2\sqrt{\kappa-4}})/({\sqrt{\kappa} + \sqrt{\kappa-4}})}.        (17)
Note that this change does not affect any of the subsequent derivations.

Figure 4 shows a comparison between numerical data and the form of the singularity for this β and κ = 16. The figure shows a good but not excellent fit to the asymptotic form.

The error mentioned here was originally detected by Panos Oikonomou, who then went on to extend the asymptotic analysis of the original article. The original invited article was devoted to the determination of the asymptotic form of the trace when there is a forcing of the form (−t)β for small negative t and β in the interval (0,1/2). The extended analysis related to negative values of β. The result of this was that the analysis of the earlier article apparently works quite well in the new regime, in which the trace asymptotes the real axis at infinity. For example, a good fit is obtained to equation (38) (without the correction term proportional to d) for β = −0.25.

The authors of this article would like to thank their colleague, Panos Oikonomou, for correcting their error and extending their work.


Dates

Issue 2 (March 2005)



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