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Values of Brownian intersection exponents. II: Plane exponents. (English) Zbl 0993.60083
[For part I, see ibid. 187, No. 2, 237–273 (2001; Zbl 1005.60097); for part III, see Ann. Inst. Henri Poincaré, Probab. Stat. 38, No. 1, 109–123 (2002; Zbl 1006.60075).]
The present paper is one in a series of papers by the authors marking a major breakthrough in the study of planar Brownian motion and beyond that in the study of a wide class of two-dimensional systems of statistical physics, including percolation and self-avoiding walks. The subject of this part of the work are the values of the intersection exponents \(\zeta\) of planar Brownian motion, which can be defined as follows: Suppose we have two families of \(n\) resp. \(m\) independent Brownian motions started in two distinct points of the unit sphere and run up to time \(t\). The probability of the union of the paths in the first family not intersecting the union of the paths in the second family is \(t^{-\zeta(n,m)+o(1)}\) as \(t\to\infty\). The main results of the paper show that \(\zeta(1,1)=5/4\) and, for \(m\geq 2\), \[ \zeta(2,m)=(1/96)((5+\sqrt{24m+1})^2-4). \]
Combining these results with those of previous papers gives a general formula, for the consistently defined intersection exponents \(\zeta(\lambda_1,\dots,\lambda_m)\) for real \(\lambda_i\geq 0\) with \(\lambda_i\geq 1\) for at least two of the arguments. Special cases of this formula have been conjectured in the physics literature for a long time.
A crucial role in the proofs is played by \(\text{SLE}_6\), the stochastic Löwner evolution process with parameter \(6\), which is conjectured to be the scaling limit of two-dimensional critical percolation cluster boundaries. The deep relation between this process, planar Brownian motion and conformal invariance is at the heart of the authors’ work and the proofs of the present paper are based on a calculation of the analogues of the exponents \(\zeta\) for this process.
Among the most interesting applications of the results of this paper (and its companions) are formulas for the Hausdorff dimension of Brownian cut points, pioneer points and of the Brownian frontier.

MSC:
60J65 Brownian motion
30C35 General theory of conformal mappings
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
60G17 Sample path properties
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