The dimension of the planar Brownian frontier is 4/3

By Gregory F. Lawler, Oded Schramm, Wendelin Werner. 2000

135 citations recorded by the source corpus.

In a series of recent preprints, we have proven that with probability one the Hausdorff dimension on the outer boundary of planar Brownian motion is 4/3, confirming a conjecture by Mandelbrot. It is also shown that the Hausdorff dimension of the set of cut points is a.s. 3/4. The present paper is an expository outline of the arguments involved in the proof of these and related results.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6e294225d2eeb839d4ef40d57abaab24187eece8

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Citations in the corpus

Liouville quantum gravity as a mating of trees (2014, background): “1) gives that the Euclidean scaling of X is x = 1 8 so that the Euclidean dimension of X should be 7 4 , in agreement with that of pioneer points of Brownian motion [LSW01a].”
The Random-Cluster Model (2002, background): “It has already provided complete explanations of conjectures, by mathematicians and physicists, associated with two-dimensional Brownian motions and specifically their intersection exponents and fractionality of frontier, [102, 103].”
Simple random walk on the uniform infinite planar quadrangulation: subdiffusivity via pioneer points (2012, methodology): “t random walks starting from the same point in a random lattice are not intersecting each other up to time nis supposed to decay as n L+o(1). 26 These exponents can be derived from the Euclidean case [30] using the KPZ relation and we have [21, Eq (3.14)] L = 1 2 L 1 2 : The special case 1 = 1=4 c”
THE SELF-AVOIDING WALK (2006, background): “In particular, Lawler, Schramm and Werner among others conjectured that SLE8/3 in a simple connected domain D $ C, provides the scaling limit for the self-avoiding walk in two dimensions (see [17] and [19] for more details).”
Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits (2001, background): “The values of various dimensions and exponents for SLE6 are known by the work of Lawler, Schramm, and Werner (see[8]), and hence similar results for percolation follow immediately.”
2D growth processes: SLE and Loewner chains (2006, background): “tioned not to touch some ﬁxed hull A is identical to the measure for hulls on D\A. This properties holds for SLE only for κ= 8/3. It also holds for Brownian excursion and this provides a way to prove [92,96] Mandelbrot’s conjecture that the fractal dimension of the exterior perimeter of Brownian exc”
Conformal loop ensembles: the Markovian characterization and the loop-soup construction (2010, background): “Note that just as for Mandelbrot’s conjecture for the dimension of Brownian boundaries [14], this statement does not involve SLE, but its proof does.”