Conformal Invariance and Intersections of Random Walks

Duplantier, Bertrand; Kwon, Ki Young

doi:10.1103/physrevlett.61.2514

Cited by 88 publications

(102 citation statements)

References 24 publications

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“…It was conjectured in [DK88b] and rigorously derived in a celebrated series of papers by Lawler, Schramm, and Werner using the Schramm-Loewner evolution with κ = 6 [LSW01a, LSW01b, LSW02]:…”

Section: Critical Liouville Quantum Gravitymentioning

confidence: 99%

Liouville quantum gravity and KPZ

2010

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Consider a bounded planar domain D, an instance h of the Gaussian free field on D, with Dirichlet energy (2π) −1 D ∇h(z) · ∇h(z)dz, and a constant 0 ≤ γ < 2. The Liouville quantum gravity measure on D is the weak limit as ε → 0 of the measureswhere dz is Lebesgue measure on D and h ε (z) denotes the mean value of h on the circle of radius ε centered at z. Given a random (or deterministic) subset X of D one can define the scaling dimension of X using either Lebesgue measure or this random measure. We derive a general quadratic relation between these two dimensions, which we view as a probabilistic formulation of the Knizhnik, Polyakov, Zamolodchikov (KPZ, 1988) relation from conformal field theory. We also present a boundary analog of KPZ (for subsets of ∂D). We discuss the connection between discrete and continuum quantum gravity and provide a framework for understanding Euclidean scaling exponents via quantum gravity. *

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Section: Critical Liouville Quantum Gravitymentioning

confidence: 99%

Liouville quantum gravity and KPZ

2010

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“…As the boundary of SLE 6 is conformally invariant, satisfies restriction, and can be well understood, computations of its exponents yielded the Brownian intersection exponents (in particular, exponents that had been predicted by Duplantier-Kwon [15,14], disconnection exponents, and Mandelbrot's conjecture [34] that the Hausdorff dimension of the boundary of planar Brownian motion is 4/3). Similarly, the determination of the critical exponents for SLE 6 in [23,24,25] combined with Smirnov's [46] proof of conformal invariance for critical percolation on the triangular lattice (along with Kesten's hyperscaling relations) facilitated proofs of several fundamental properties of critical percolation [47,28,45], some of which had been predicted in the theoretical physics literature, e.g., [37,35,36,38,43].…”

Section: Introductionmentioning

confidence: 99%

Conformal restriction: The chordal case

Lawler

Schramm

Werner³

2003

J. Amer. Math. Soc.

237

554

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We characterize and describe all random subsets K of a given simply connected planar domain (the upper half-plane H, say) which satisfy the "conformal restriction" property, i.e., K connects two fixed boundary points (0 and ∞, say) and the law of K conditioned to remain in a simply connected open subset H of H is identical to that of Φ(K), where Φ is a conformal map from H onto H with Φ(0) = 0 and Φ(∞) = ∞. The construction of this family relies on the stochastic Loewner evolution processes with parameter κ ≤ 8/3 and on their distortion under conformal maps. We show in particular that SLE 8/3 is the only random simple curve satisfying conformal restriction and relate it to the outer boundaries of planar Brownian motion and SLE 6 .

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“…This result had been predicted by Duplantier-Kwon [15], see also [13,14] for another non-rigorous derivation based on quantum gravity ideas and predictions by Khnizhnik, Polyakov and Zamolodchikov.…”

Section: Intersection Exponentsmentioning

confidence: 64%