Asymptotic behaviour of disconnection and non-intersection exponents (Q1355718)

The probability that \(n\) independent planar Brownian paths started away from 0 up to time \(t\) do not disconnect 0 from infinity is logarithmically equivalent to \(t^{-\eta_n/2}\), where \(\eta_n\) is called the disconnection exponent for \(n\) paths. It is proved that \(\lim_{n\to\infty}\eta_n/n=1/2\). Another interesting exponent is the disconnection exponent \(\xi(n,p)\) for \(n\) paths versus \(p\) paths; using conformal invariance and introducing spiders and colourings, the author proves that for \(a>0\), \(b>0\), \(\lim_{n\to\infty} \xi([na], [nb])/n= (\sqrt{a}+\sqrt{b})^2/2\).