zbMATH — the first resource for mathematics

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.

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
Full Text: DOI arXiv
[1] Ahlfors, L. V.,Conformal Invariants: Topics in Geometric Function Theory. McGraw-Hill, New York, 1973. · Zbl 0272.30012
[2] Aizenman, M., Duplantier, B. &Aharony, A., Path crossing exponents and the external perimeter in 2D percolation.Phys. Rev. Lett., 83 (1999), 1359–1362.
[3] Azencott, R., Behaviour of diffusion semi-groups at infinity.Bull. Soc. Math. France, 102 (1974), 193–240. · Zbl 0293.60071
[4] Bishop, C. J., Jones, P. W., Pemantle, R. &Peres, Y., The dimension of the Brownian frontier is greater than 1.J. Funct. Anal., 143 (1997), 309–336. · Zbl 0870.60077
[5] Burdzy, K. &Lawler, G. F., Non-intersection exponents for Brownian paths. Part I: Existence and an invariance principle.Probab. Theory Related Fields, 84 (1990), 393–410. · Zbl 0685.60080
[6] –, Non-intersection exponents for Brownian paths. Part II: Estimates and applications to a random fractal.Ann. Probab., 18 (1990), 981–1009. · Zbl 0719.60085
[7] Cardy, J. L., Conformal invariance and surface critical behavior.Nuclear Phys. B, 240 (1984), 514–532.
[8] –, Critical percolation in finite geometries.J. Phys. A, 25 (1992), L201-L206. · Zbl 0965.82501
[9] –, The number of incipient spanning clusters in two-dimensional percolation.J. Phys. A, 31 (1998), L105. · Zbl 0973.82021
[10] Cranston, M. &Mountford, T., An extension of a result of Burdzy and Lawler.Probab. Theory Related Fields, 89 (1991), 487–502. · Zbl 0725.60072
[11] Duplantier, B., Random walks and quantum gravity in two dimensions.Phys. Rev. Lett., 81 (1998), 5489–5492. · Zbl 0949.83056
[12] –, Two-dimensional copolymers and exact conformal multifractality.Phys. Rev. Lett., 82 (1999), 880–883.
[13] –, Harmonic measure exponents for two-dimensional percolation.Phys. Rev. Lett., 82 (1999), 3940–3943. · Zbl 1042.82560
[14] Duplantier, B. &Kwon, K.-H., Conformal invariance and intersection of random walks.Phys. Rev. Lett., 61 (1988), 2514–2517.
[15] Duplantier, B. &Saleur, H., Exact determination of the percolation hull exponent in two dimensions.Phys. Rev. Lett., 58 (1987), 2325–2328.
[16] Kenyon, R., Conformal invariance of domino tiling.Ann. Probab., 28 (2000), 759–795. · Zbl 1043.52014
[17] –, Long-range properties of spanning trees. Probabilistic techniques in equilibrium and nonequilibrium statistical physics.J. Math. Phys., 41 (2000), 1338–1363. · Zbl 0977.82011
[18] –, The asymptotic determinant of the discrete Laplacian.Acta Math., 185 (2000), 239–286. · Zbl 0982.05013
[19] Lawler, G. F.,Intersections of Random Walks. Birkhäuser Boston, Boston, MA, 1991. · Zbl 1228.60004
[20] –, Hausdorff dimension of cut points for Brownian motion.Electron. J. Probab., 1:2 (1996), 1–20 (electronic). · Zbl 0891.60078
[21] –, Cut times for simple random walk.Electron. J. Probab., 1:13 (1996), 1–24 (electronic). · Zbl 0888.60059
[22] –, The dimension of the frontier of planar Brownian motion.Electron. Comm. Probab., 1:5 (1996), 29–47 (electronic). · Zbl 0857.60083
[23] Lawler, G. F., The frontier of a Brownian path is multifractal. Preprint, 1997.
[24] –, Strict concavity of the intersection exponent for Brownian motion in two and three dimensions.Math. Phys. Electron. J., 4:5 (1998), 1–67 (electronic). · Zbl 0909.60065
[25] –, Geometric and fractal properties of Brownian motion and random walk paths in two and three dimensions, inRandom Walks (Budapest, 1998), pp. 219–258. Bolyai Soc. Math. Stud., 9. Janos Bolyai Math. Soc., Budapest, 1999.
[26] Lawler, G. F. &Puckette, E. E., The intersection exponent for simple random walk.Combin. Probab. Comput., 9 (2000), 441–464. · Zbl 0974.60088
[27] Lawler, G. F., Schramm, O. &Werner, W., Values of Brownian intersection exponents, I: Half-plane exponents.Acta Math., 187 (2001), 237–273. · Zbl 1005.60097
[28] Lawler, G. F., Schramm, O. & Werner, W., Values of Brownian intersection exponents, III: Two-sided exponents. To appear inAnn. Inst. H. Poincaré Probab. Statist. http://arxiv.org/abs/math.PR/0005294. · Zbl 1006.60075
[29] Lawler, G. F., Schramm, O. & Werner, W., Analyticity of intersection exponents for planar Brownian motion. To appear inActa Math., 188 (2002). http://arxiv.org/abs/math.PR/0005295. · Zbl 1024.60033
[30] Lawler, G. F., Schramm, O. & Werner, W., Sharp estimates for Brownian non-intersection probabilities. To appear inIn and Out of Equilibrium. Probability with a Physics Flavor. Progr. Probab. Birkhäuser Boston, Boston, MA.
[31] Lawler, G. F. &Werner, W., Intersection exponents for planar Brownian motion.Ann. Probab., 27 (1999), 1601–1642. · Zbl 0965.60071
[32] –, Universality for conformally invariant intersection exponents.J. Eur. Math. Soc. (JEMS), 2 (2000), 291–328. · Zbl 1098.60081
[33] Löwner, K., Untersuchungen über schlichte konforme Abbildungen des Einheitskreises, I.Math. Ann., 89 (1923), 103–121. · JFM 49.0714.01
[34] Madras, N. &Slade, G.,The Self-Avoiding Walk. Birkhäuser Boston, Boston, MA, 1993. · Zbl 0780.60103
[35] Mandelbrot, B. B.,The Fractal Geometry of Nature. Freeman, San Francisco, CA, 1982. · Zbl 0504.28001
[36] Nienhuis, B., Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas.J. Statist. Phys., 34 (1984), 731–761. · Zbl 0595.76071
[37] Pommerenke, Ch., On the Loewner differential equation.Michigan Math. J., 13 (1966), 435–443. · Zbl 0163.31801
[38] Revuz, D. &Yor, M.,Continuous Martingales and Brownian Motion. Grundlehren Math. Wiss., 293. Springer-Verlag, Berlin, 1991. · Zbl 0731.60002
[39] Schramm, O., Scaling limits of loop-erased random walks and uniform spanning trees.Israel J. Math., 118 (2000), 221–288. · Zbl 0968.60093
[40] Werner, W., Bounds for disconnection exponents.Electron. Comm. Probab., 1:4 (1996), 19–28 (electronic). · Zbl 0862.60069
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.