By Gregory F. Lawler

Theoretical physicists have envisioned that the scaling limits of many two-dimensional lattice versions in statistical physics are in a few feel conformally invariant. This trust has allowed physicists to foretell many amounts for those serious platforms. the character of those scaling limits has lately been defined accurately by utilizing one recognized software, Brownian movement, and a brand new building, the Schramm-Loewner evolution (SLE). This booklet is an advent to the conformally invariant techniques that seem as scaling limits. the subsequent subject matters are coated: stochastic integration; advanced Brownian movement and measures derived from Brownian movement; conformal mappings and univalent features; the Loewner differential equation and Loewner chains; the Schramm-Loewner evolution (SLE), that is a Loewner chain with a Brownian movement enter; and purposes to intersection exponents for Brownian movement. the necessities are first-year graduate classes in genuine research, advanced research, and chance. The publication is appropriate for graduate scholars and examine mathematicians attracted to random strategies and their purposes in theoretical physics.

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P(t, z, w) dt = 2πt π 0 0 Harmonicity of G D (z, ·) follows from 1 ∆w GD (z, w) = 2 ∞ p˙ D (t, z, w) dt 0 and pD (0+, z, w) = pD (∞, z, w) = 0. Alternatively, one can show that it satisfies the mean value property. To show uniqueness, suppose φ is a harmonic function in D \ {z} such that φ is continuous on D \ {z}; φ ≡ 0 on ∂D and as w → z, φ(w) = − log |w − z| + O(1). Let Dz,ǫ = D \ B(z, ǫ). 11 to Dz,ǫ , we get φ(w) = Pw {τDz,ǫ < τD } [− log ǫ + O(1)]. By letting ǫ → 0+, we get that φ(w) = lim −(log ǫ) Pw {τDz,ǫ < τD }.

10. Suppose D is a domain such that ∂D has at least one regular point, and F : ∂D → R is a bounded, measurable function. 1) u(z) = Ez [F (BτD )], z ∈ D. Then u is a bounded, harmonic function in D and is continuous at all regular points z ∈ ∂D at which F is continuous. Proof. It is obvious that u ∞ = F ∞ < ∞. The rotational symmetry of Brownian motion and the strong Markov property show that u satisfies the mean value property in D and hence is harmonic in D. 4. 11. Suppose D is a regular domain, and F : ∂D → R is a bounded, continuous function.

Then d′ 4 d′ ≤ |f ′ (z)| ≤ , 4d d where d = dist(z, ∂D), d′ = dist(z ′ , ∂D′ ). Proof. Assume without loss of generality that z = z ′ = 0. Then, f(dw) f˜(w) := ∈ S. d f ′ (0) By the Koebe 1/4 Theorem, B(0, 1/4) ⊂ f˜(D ), and hence d′ ≥ (1/4) |f ′ (0)| d. This gives the upper bound, and the lower bound is obtained by considering f −1 . 19. If f ∈ S and z ∈ D , 4 |z| 2 |z|2 z f ′′ (z) ≤ − . ′ 2 f (z) 1 − |z| 1 − |z|2 obius transformation with Proof. Let Tz (w) = (w + z)/(1 + zw) be the M¨ Tz (0) = z and Tz′ (0) = 1 − |z|2.