Abstract
The study of conformal restriction properties in two-dimensions has been initiated by Lawler et al. (J Am Math Soc 16(4):917–955, 2003) who focused on the natural and important chordal case: they characterized and constructed all random subsets of a given simply connected domain that join two marked boundary points and that satisfy the additional restriction property. The radial case (sets joining an inside point to a boundary point) has then been investigated by Wu (Stoch Process Appl 125(2):552–570, 2015). In the present paper, we study the third natural instance of such restriction properties, namely the “trichordal case”, where one looks at random sets that join three marked boundary points. This case involves somewhat more technicalities than the other two, as the construction of this family of random sets relies on special variants of SLE\(_{8/3}\) processes with a drift term in the driving function that involves hypergeometric functions. It turns out that such a random set can not be a simple curve simultaneously in the neighborhood of all three marked points, and that the exponent \(\alpha = 20/27\) shows up in the description of the law of the skinniest possible symmetric random set with this trichordal restriction property.
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Acknowledgements
The author is very grateful to Wendelin Werner for suggesting this question, for numerous discussions and suggestions, and for his help throughout the preparation of this paper. The author acknowledges support of the SNF Grant SNF-155922. The author is also part of the NCCR Swissmap.
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Appendix: Itô calculus for the proof of Lemma 5.3
Appendix: Itô calculus for the proof of Lemma 5.3
We need to show that the process
is a local martingale, where \((W_t, O_t, V_t)\) satisfy
This looks like a daunting task, but it is a rather straightforward direct Itô formula computation: to simplify the notations, set
First calculate the following Itô derivatives:
Then we have
From the formula above, we see that the local martingale term of \(d M_t/ M_t\) is \(L_t d B_t\) (and therefore \(d\langle M \rangle _t =M_t^2 L_t^2 dt\)), where
The drift term of \(d M_t/ M_t\) can be written as \(D_t dt\) where
Recall that by (4.17) we have
In \(D_t\), we now interpret the terms that only involve \(W_t, O_t, V_t\) as ‘constant’ and we regroup the coefficients in front of the terms which involve \(h_t\). The coefficients for each of these terms are as follows: the ‘constant’ term is
and the remaining ones can be listed as follows:
Adding up the terms in (A.4–A.9), we get
where
As G satisfies the differential Eq. (4.12), we can conclude that \(E_t=0\). Now, it remains only to deal with the ‘constant’ term. It is equal to \(C_t / ( ( V_t-W_t)(O_t-W_t)) \), with
which is also equal to 0 because of (4.12).
This finally proves that \(D_t=0\) hence \(M_t\) is a local martingale.
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Qian, W. Conformal restriction: the trichordal case. Probab. Theory Relat. Fields 171, 709–774 (2018). https://doi.org/10.1007/s00440-017-0791-z
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DOI: https://doi.org/10.1007/s00440-017-0791-z
Mathematics Subject Classification
- Primary 60D05
- 60J67
- Secondary 60K35
- 30C99