Abstract
We define the Schramm–Loewner evolutions in this chapter and study their basic properties.
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Notes
- 1.
Oded Schramm (1961–2008) was an Israeli-American mathematician, who was a highly influential researcher in the fields of complex analysis and probability theory and is best known for inventing SLE and deriving many of its properties (with his co-authors) as well as many other insightful results around random processes related to statistical physics. He died tragically in a climbing accident.
- 2.
Technically here we should restrict to so called stopping times.
- 3.
By an argument which we leave as an exercise, when CI and DMP are satisfied, the half-plane capacity of the hull \(\gamma [0,t]\) will tend to infinity as t tends to infinity. Therefore the reparameterized curve will be parametrized by the set \(\mathbb {R}_{\ge 0}\).
- 4.
We use here a variant of the notation so that \(W_t,K_t,g_t(z)\) etc. are replaced by W(t), K(t), g(t, z).
- 5.
Remember that denotes equal in distribution.
- 6.
We will prove the theorem only for \(\kappa \ne 8\). The case \(\kappa =8\) is a consequence of the results of [9].
- 7.
The s-dimensional Hausdorff measure \(\mathscr {H}^s\) is defined by
$$\begin{aligned} \mathscr {H}^s[\varGamma ] = \lim _{\delta \rightarrow 0} \;\inf \left\{ \sum _{k=1}^\infty ({{\mathrm{diam}}}(V_k))^s \,:\, \varGamma \subset \bigcup _{k=1}^\infty V_k \text { and } {{\mathrm{diam}}}(V_k)<\delta \right\} \end{aligned}$$where the infimum is over all countable covers \(V_k\), \(k=1,2,\ldots , n\), of \(\varGamma \) satisfying \( {{\mathrm{diam}}}(V_k)<\delta \) for all k.
- 8.
It is convenient to use complex valued Itô processes. It is understood that an equality of the form \(\mathrm {d}Z(t) = \xi (t) \mathrm {d}t + \sum _{k=1}^n \zeta _k(t) \mathrm {d}B_k(t)\), where \((B_k(t))_{t \in \mathbb {R}_{\ge 0}}\) are standard one-dimensional Brownian motions, means that the real and imaginary part of both of the sides are equal when we consider \(\mathrm {d}t\) and \(\mathrm {d}B_k(t)\) to be real.
- 9.
The existence and uniqueness of the solution follows from Theorem 2.9, for instance, using the following trick. For any \(n \in \mathbb {N}\), replace the drift term by a smooth continuation of the function that maps \(x \mapsto (\delta -1)/(2 x)\), \(x>1/n\) and \(x \mapsto 0\), \(x<0\). The drift term and the approximating drift term are identical on the interval \([1/n, +\infty )\) and thus the solutions agree until the process exits the interval.
- 10.
Conformally covariant here means that the transformation rule under conformal maps is simple.
- 11.
Here and below \(z^\alpha \) is defined as \(e^{\alpha \log z}\) where the branch of \(\log \) is such that \({\text {Im}}\log z \in [0,\pi ]\) for \(z \in \overline{\mathbb {H}}\).
References
Beffara, V.: The dimension of the SLE curves. Ann. Probab. 36(4), 1421–1452 (2008). https://doi.org/10.1214/07-AOP364
Durrett, R.: Probability: theory and examples, 4th edn. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (2010). https://doi.org/10.1017/CBO9780511779398
Hongler, C., Kytölä, K.: Ising interfaces and free boundary conditions. J. Am. Math. Soc. 26(4), 1107–1189 (2013). https://doi.org/10.1090/S0894-0347-2013-00774-2
Kemppainen, A.: Stationarity of SLE. J. Stat. Phys. 139(1), 108–121 (2010). https://doi.org/10.1007/s10955-010-9929-4
Lawler, G., Schramm, O., Werner, W.: Conformal restriction: the chordal case. J. Am. Math. Soc. 16(4), 917–955 (electronic) (2003). https://doi.org/10.1090/S0894-0347-03-00430-2
Lawler, G.F.: Conformally Invariant Processes in the Plane, Mathematical Surveys and Monographs, vol. 114. Am. Math. Soc., Providence, RI (2005)
Lawler, G.F.: Schramm-Loewner Evolution (SLE). In: Statistical mechanics, IAS/Park City Math. Ser., vol. 16, pp. 231–295. Amer. Math. Soc., Providence, RI (2009)
Lawler, G.F.: Fractal and Multifractal Properties of Schramm-Loewner Evolution. In: Probability and statistical physics in two and more dimensions, Clay Math. Proc., vol. 15, pp. 277–318. Amer. Math. Soc., Providence, RI (2012)
Lawler, G.F., Schramm, O., Werner, W.: Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32(1B), 939–995 (2004). https://doi.org/10.1214/aop/1079021469
Rohde, S., Schramm, O.: Basic properties of SLE. Ann. of Math.(2) 161(2), 883–924 (2005). https://doi.org/10.4007/annals.2005.161.883
Schramm, O.: Scaling limits of loop-erased random walks and uniform spanning trees. Isr. J. Math. 118, 221–288 (2000). https://doi.org/10.1007/BF02803524
Werner, W.: Random planar curves and Schramm-Loewner evolutions. In: Lectures on probability theory and statistics, Lecture Notes in Math., vol. 1840, pp. 107–195. Springer, Berlin (2004). https://doi.org/10.1007/978-3-540-39982-7_2
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Kemppainen, A. (2017). Schramm–Loewner Evolution. In: Schramm–Loewner Evolution. SpringerBriefs in Mathematical Physics, vol 24. Springer, Cham. https://doi.org/10.1007/978-3-319-65329-7_5
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