Abstract
In this introductory chapter, we look at iterations of conformal maps, random processes, such as random walks, and statistical physics and establish some connections.
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Notes
- 1.
- 2.
In the sense that if \(P_1\) and \(P_2\) are smooth curves that form an angle \(\theta \) at \(z_0\), then also \(f \circ P_1\) and \(f \circ P_2\) form an angle \(\theta \) at \(f(z_0)\).
- 3.
Throughout this text we use the notations \(\mathbb {Z}_{>0}= \{k \in \mathbb {Z}\,:\, k \ge 1\}\), \(\mathbb {Z}_{\ge 0}= \{k \in \mathbb {Z}\,:\, k \ge 0\}\), \(\mathbb {R}_{>0}= \{x \in \mathbb {R}\,:\, x > 0\}\), \(\mathbb {R}_{\ge 0}= \{x \in \mathbb {R}\,:\, x \ge 0\}\) as well as \(\llbracket j, k\rrbracket \) for the ordered set \(j,j+1,j+2,\ldots , k-1,k\), where \(j<k\) are integers.
- 4.
We use a common notion that \(\lfloor x \rfloor \) is the largest integer smaller or equal to x.
- 5.
Such a limit is an example of scaling limit . Two typical features of a scaling limit are that there are scaling factor involved, such as \(n^{-a}\) and \(n^a\) above, which ensure that the limit exists, and that the limiting object will be described by continuous variables (another term is a continuum limit).
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- 7.
Remember that, in this particular case, the random variables, \(X_k\), \(k \in \mathbb {Z}_{>0}\), are independent if for any \(n \in \mathbb {Z}_{>0}\) and for any \(x_1,x_2,\ldots ,x_n \in \{-1,+1\}\), \(\mathsf {P}[\,X_k = x_k \text { for all } k\in \llbracket 1,n\rrbracket \,]=\prod _{ k\in \llbracket 1,n\rrbracket } \mathsf {P}[X_k = x_k] \).
- 8.
We use the notation \((X_t)_{t \in I}\) where usually \(I=\mathbb {Z}_{\ge 0}\) or \(I=\mathbb {R}_{\ge 0}\), to denote a stochastic process.
- 9.
Remember that the result that a sum of independent and identical centered random variables scaled by \(n^{-1/2}\) converges to a Gaussian random variable in distribution, is called the central limit theorem.
- 10.
Simply connectedness means that the domain consisting of the hexagons is a simply connected domain (i.e. with no holes)—in other words, if we have a closed path of hexagons in the domain, it cannot disconnect any point in the complement of the domain from infinity.
- 11.
From the modelling perspective, the open sites represent channels through which a substance, say, water can flow. Therefore if we inject water into the sites of a set \(A_1\), the water will flow to all the sites connected by a path of open sites to \(A_1\). In particular we are interested in connection events that for fixed \(A_1\) and \(A_2\) there exists a connected path from \(A_1\) to \(A_2\) that stays in a set B.
- 12.
The reader should stop to think this for a moment, though.
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Kemppainen, A. (2017). Introduction. In: Schramm–Loewner Evolution. SpringerBriefs in Mathematical Physics, vol 24. Springer, Cham. https://doi.org/10.1007/978-3-319-65329-7_1
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