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Conformal restriction: the trichordal case

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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References

  1. Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications Inc., New York (1992). (Reprint of the 1972 edition)

    MATH  Google Scholar 

  2. Bauer, M., Bernard, D., Kytölä, K.: Multiple Schramm–Loewner evolutions and statistical mechanics martingales. J. Stat. Phys. 120(5–6), 1125–1163 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Dubédat, J.: \({\rm SLE}(\kappa,\rho )\) martingales and duality. Ann. Probab. 33(1), 223–243 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dubédat, J.: Euler integrals for commuting SLEs. J. Stat. Phys. 123(6), 1183–1218 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dubédat, J.: Commutation relations for Schramm–Loewner evolutions. Commun. Pure Appl. Math. 60(12), 1792–1847 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Duplantier, B.: Random walks and quantum gravity in two dimensions. Phys. Rev. Lett. 81(25), 5489–5492 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Duplantier, B.: Conformal fractal geometry and boundary quantum gravity. In: Fractal Geometry and Applications: a Jubilee of Benoît Mandelbrot, Part 2, volume 72 of Proc. Sympos. Pure Math., pp. 365–482. Amer. Math. Soc., Providence (2004)

  8. Lawler, G.F.: Conformally Invariant Processes in the Plane. Mathematical Surveys and Monographs, vol. 114. American Mathematical Society, Providence (2005)

    Google Scholar 

  9. Lawler, G.F.: Partition functions, loop measure, and versions of SLE. J. Stat. Phys. 134(5–6), 813–837 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Lawler, G.F., Schramm, O., Werner, W.: Values of Brownian intersection exponents. I. Half-plane exponents. Acta Math. 187(2), 237–273 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lawler, G.F., Schramm, O., Werner, W.: Values of Brownian intersection exponents. II. Plane exponents. Acta Math. 187(2), 275–308 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lawler, G.F., Schramm, O., Werner, W.: Values of Brownian intersection exponents. III. Two-sided exponents. Ann. Inst. H. Poincaré Probab. Stat. 38(1), 109–123 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lawler, G.F., Schramm, O., Werner, W.: Conformal restriction: the chordal case (electronic). J. Am. Math. Soc. 16(4), 917–955 (2003)

    Article  MATH  Google Scholar 

  14. Lawler, G.F., Werner, W.: Intersection exponents for planar Brownian motion. Ann. Probab. 27(4), 1601–1642 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  15. Miller, J., Sheffield, S.: Imaginary geometry I: interacting SLEs. Probab. Theory Related Fields 164(3–4), 553–705 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Rohde, S., Schramm, O.: Basic properties of SLE. Ann. Math. (2) 161(2), 883–924 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Schramm, O.: Scaling limits of loop-erased random walks and uniform spanning trees. Isr. J. Math. 118, 221–288 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Schramm, O., Wilson, D.B.: SLE coordinate changes (electronic). N. Y. J. Math. 11, 659–669 (2005)

    MATH  Google Scholar 

  19. Werner, W.: Girsanov’s transformation for \(\text{ SLE }(\kappa,\rho )\) processes, intersection exponents and hiding exponents. Ann. Fac. Sci. Toulouse Math. (6) 13(1), 121–147 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Werner, W.: Conformal restriction and related questions. Probab. Surv. 2, 145–190 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wu, H.: Conformal restriction: the radial case. Stoch. Process. Appl. 125(2), 552–570 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhan, D.: Reversibility of some chordal \({\rm SLE}(\kappa; \rho )\) traces. J. Stat. Phys. 139(6), 1013–1032 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

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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Correspondence to Wei Qian.

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

$$\begin{aligned} M_t=h_t'(W_t)^{5/8}h_t'(O_t)^{n}h_t'(V_t)^m \left( \frac{h_t(O_t)-h_t(V_t)}{O_t-V_t}\right) ^{l-m-n+\lambda } {G\left( \frac{h_t(W_t)-h_t(O_t)}{h_t(O_t)-h_t(V_t)}\right) }\Biggm / {G\left( \frac{W_t-O_t}{O_t-V_t}\right) } \end{aligned}$$

is a local martingale, where \((W_t, O_t, V_t)\) satisfy

$$\begin{aligned} \left\{ \begin{aligned}&\; dO_t=\frac{2dt}{O_t-W_t}, \quad dV_t=\frac{2dt}{V_t-W_t},\\&\; dW_t=\sqrt{\kappa }dB_t+J(W_t-O_t,W_t-V_t)dt. \\ \end{aligned} \right. \end{aligned}$$

This looks like a daunting task, but it is a rather straightforward direct Itô formula computation: to simplify the notations, set

$$\begin{aligned} {\tilde{X}}_t := \frac{h_t(W_t)-h_t(O_t)}{h_t(O_t)-h_t(V_t)},\quad X_t := \frac{W_t-O_t}{O_t-V_t}, \quad \hbox {and} \quad J_t := J(W_t-O_t,W_t-V_t). \end{aligned}$$

First calculate the following Itô derivatives:

$$\begin{aligned}&d\, h_t(W_t)= \left[ -\frac{5}{3} h_t''(W_t)+h_t'(W_t) J_t \right] dt +\sqrt{\frac{8}{3}} h_t'(W_t) d B_t\\&d\, h_t'(W_t) =\left[ \frac{h_t''(W_t)^2}{2h_t'(W_t)} +h_t''(W_t) J_t \right] dt +\sqrt{\frac{8}{3}} h_t''(W_t) d B_t\\&d\, h_t(O_t)=\frac{2h_t'(W_t)^2}{h_t(O_t)-h_t(W_t)} dt, \quad h_t(V_t)=\frac{2h_t'(W_t)^2}{h_t(V_t)-h_t(W_t)} dt\\&d\, h_t'(O_t)=\left[ -\frac{2h_t'(W_t)^2 h_t'(O_t)}{(h_t(O_t)-h_t(W_t))^2} +\frac{2h_t'(O_t)}{(O_t-W_t)^2} \right] dt\\&d\, h_t'(V_t)=\left[ -\frac{2h_t'(W_t)^2 h_t'(V_t)}{(h_t(V_t)-h_t(W_t))^2} +\frac{2h_t'(V_t)}{(V_t-W_t)^2} \right] dt. \end{aligned}$$

Then we have

$$\begin{aligned} \frac{d M_t}{M_t}= & {} d\, \log M_t +\frac{1}{2M_t^2} d\langle M\rangle _t\\= & {} \frac{5}{8} \frac{d h_t'(W_t)}{h_t'(W_t)} -\frac{5h_t''(W_t)^2}{6 h_t'(W_t)^2} dt + n \frac{dh_t'(O_t)}{h_t'(O_t)} +m \frac{dh_t'(V_t)}{h_t'(V_t)} \\&+ (l-m-n+\lambda ) \frac{d h_t(O_t)-d h_t(V_t) }{{h_t(O_t)-h_t(V_t)}} -(l-m-n+\lambda ) \frac{d O_t -d V_t}{{O_t-V_t}} \\&+ \frac{d\,{G\left( {\tilde{X}}_t\right) }}{G\left( {\tilde{X}}_t\right) } -\frac{4G'\left( {\tilde{X}}_t\right) ^2}{3G\left( {\tilde{X}}_t\right) ^2} \left( \frac{h_t'(W_t)}{h_t(O_t)-h_t(V_t)} \right) ^2 dt\\&- \frac{d\, {G(X_t)}}{G(X_t)} +\frac{4G'(X_t)^2}{3G(X_t)^2}\frac{1}{\left( O_t-V_t \right) ^2} dt + \frac{1}{2M_t^2} d\langle M\rangle _t. \end{aligned}$$

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

$$\begin{aligned} L_t=\frac{5}{8} \sqrt{\frac{8}{3}}\frac{h_t''(W_t)}{h_t'(W_t)} +\frac{G'\left( {\tilde{X}}_t\right) }{G\left( {\tilde{X}}_t\right) } \sqrt{\frac{8}{3}} \frac{h_t'(W_t)}{h_t(O_t)- h_t(V_t)} -\frac{G'(X_t)}{G(X_t)}\sqrt{\frac{8}{3}} \frac{1}{O_t-V_t}. \end{aligned}$$

The drift term of \(d M_t/ M_t\) can be written as \(D_t dt\) where

$$\begin{aligned} D_t&=\frac{5}{8} \left[ \frac{h_t''(W_t)^2}{2h_t'(W_t)^2} +\frac{h_t''(W_t)}{h_t'(W_t)} J_t \right] -\frac{5h_t''(W_t)^2}{6 h_t'(W_t)^2} \\&\quad +n \left[ -\frac{2h_t'(W_t)^2 }{(h_t(O_t)-h_t(W_t))^2} +\frac{2}{(O_t-W_t)^2} \right] \\&\quad + m \left[ -\frac{2h_t'(W_t)^2 }{(h_t(V_t)-h_t(W_t))^2} +\frac{2}{(V_t-W_t)^2} \right] \\&\quad +(l-m-n+\lambda ) \frac{-2 h_t'(W_t)^2 }{\left( h_t(O_t)-h_t(W_t) \right) \left( h_t(V_t)-h_t(W_t) \right) }\\&\quad -(l-m-n+\lambda ) \frac{-2}{(O_t-W_t)(V_t-W_t)}\\&\quad + \frac{G'\left( {\tilde{X}}_t\right) }{G\left( {\tilde{X}}_t\right) }\cdot \frac{1}{h_t(O_t)-h_t(V_t)} \left[ \left( -\frac{5}{3} h_t''(W_t)+h_t'(W_t) J_t \right) - \frac{2h_t'(W_t)^2}{h_t(O_t)-h_t(W_t)} \right] \\&\quad -\frac{G'\left( {\tilde{X}}_t\right) }{G\left( {\tilde{X}}_t\right) }\cdot \frac{h_t(W_t)-h_t(O_t)}{\left( h_t(O_t)-h_t(V_t)\right) ^2} 2 h_t'(W_t)^2 \frac{h_t(V_t)-h_t(O_t)}{\left( h_t(O_t)-h_t(W_t) \right) \left( h_t(V_t)-h_t(W_t) \right) }\\&\quad +\frac{G''\left( {\tilde{X}}_t\right) }{2G\left( {\tilde{X}}_t\right) }\cdot \frac{8}{3} \frac{h_t'(W_t)^2}{\left( h_t(O_t)- h_t(V_t) \right) ^2} -\frac{4G'\left( {\tilde{X}}_t\right) ^2}{3G\left( {\tilde{X}}_t\right) ^2} \left( \frac{h_t'(W_t)}{h_t(O_t)-h_t(V_t)} \right) ^2\\&\quad -\frac{G'(X_t)}{G(X_t)}\cdot \left( \frac{J_t}{O_t-V_t}-\frac{2}{(O_t-W_t)(O_t-V_t)} \right) \\&\quad +\frac{G'(X_t)}{G(X_t)}\cdot \frac{W_t-O_t}{(O_t-V_t)^2} \cdot \frac{2(V_t-O_t)}{(O_t-W_t)(V_t-W_t)}\\&\quad -\frac{G''(X_t)}{2G(X_t)} \cdot \frac{8}{3} \frac{1}{(O_t-V_t)^2} +\frac{4G'(X_t)^2}{3G(X_t)^2}\frac{1}{\left( O_t-V_t \right) ^2} +\frac{1}{2} L_t^2. \end{aligned}$$

Recall that by (4.17) we have

$$\begin{aligned} J_t=\frac{8}{3}\frac{G'(X_t)}{G( X_t)}\frac{1}{O_t-V_t}. \end{aligned}$$
(A.1)

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

$$\begin{aligned}&\frac{2n}{(O_t-W_t)^2} +\frac{2m}{(V_t-W_t)^2} +2\frac{l-m-n+\lambda }{(O_t-W_t)(V_t-W_t)}\\&\quad -\frac{G'(X_t)}{G(X_t)}\cdot \left( \frac{J_t}{O_t-V_t}-\frac{2}{(O_t-W_t)(O_t-V_t)} \right) \\&\quad +\frac{G'(X_t)}{G(X_t)}\cdot \frac{W_t-O_t}{(O_t-V_t)^2} \cdot \frac{2(V_t-O_t)}{(O_t-W_t)(V_t-W_t)} -\frac{G''(X_t)}{2G(X_t)} \cdot \frac{8}{3} \frac{1}{(O_t-V_t)^2} \\&\quad +\frac{4}{3}\frac{G'(X_t)^2}{G( X_t)^2}\frac{1}{(O_t-V_t)^2} +\frac{4G'(X_t)^2}{3G(X_t)^2}\frac{1}{\left( O_t-V_t \right) ^2} \end{aligned}$$

and the remaining ones can be listed as follows:

$$\begin{aligned} \frac{h_t''(W_t)^2}{h_t'(W_t)^2}: \quad&\frac{5}{16}-\frac{5}{6} +\frac{25}{48}=0 \end{aligned}$$
(A.2)
$$\begin{aligned} \frac{h_t''(W_t)}{h_t'(W_t)}: \quad&\frac{5}{8} J_t-\frac{5}{3} \frac{G'(X_t)}{G(X_t)}\cdot \frac{1}{O_t-V_t}=0 \end{aligned}$$
(A.3)
$$\begin{aligned} \frac{h_t'(W_t)^2 }{(h_t(O_t)-h_t(W_t))^2} : \quad&-2n \end{aligned}$$
(A.4)
$$\begin{aligned} \frac{h_t'(W_t)^2 }{(h_t(V_t)-h_t(W_t))^2} : \quad&-2m \end{aligned}$$
(A.5)
$$\begin{aligned} \frac{h_t'(W_t)^2 }{(h_t(O_t)-h_t(V_t))^2} : \quad&\frac{4}{3} \frac{G''\left( {\tilde{X}}_t\right) }{G\left( {\tilde{X}}_t\right) } +\frac{4}{3} \frac{G'\left( {\tilde{X}}_t\right) ^2}{G\left( {\tilde{X}}_t\right) ^2}-\frac{4G'\left( {\tilde{X}}_t\right) ^2}{3G\left( {\tilde{X}}_t\right) ^2}\nonumber \\&= \frac{4}{3} \frac{G''\left( {\tilde{X}}_t\right) }{G\left( {\tilde{X}}_t\right) } \end{aligned}$$
(A.6)
$$\begin{aligned} \frac{h_t'(W_t)^2}{\left( h_t(O_t)-h_t(W_t) \right) \left( h_t(V_t)-h_t(W_t) \right) }:\quad&-2(l-m-n+\lambda ) \end{aligned}$$
(A.7)
$$\begin{aligned} \frac{h_t'(W_t)^2}{\left( h_t(O_t)-h_t(W_t)\right) \left( h_t(O_t)-h_t(V_t) \right) }: \quad&-2\frac{G'\left( {\tilde{X}}_t\right) }{G\left( {\tilde{X}}_t\right) } \end{aligned}$$
(A.8)
$$\begin{aligned} \frac{h_t'(W_t)^2}{\left( h_t(V_t)-h_t(W_t)\right) \left( h_t(O_t)-h_t(V_t) \right) }: \quad&-2\frac{G'\left( {\tilde{X}}_t\right) }{G\left( {\tilde{X}}_t\right) } \end{aligned}$$
(A.9)
$$\begin{aligned} \frac{h_t''(W_t)}{h_t(O_t)-h_t(V_t)}: \quad&-\frac{5G'\left( {\tilde{X}}_t\right) }{3G\left( {\tilde{X}}_t\right) }+ \frac{5G'\left( {\tilde{X}}_t\right) }{3G\left( {\tilde{X}}_t\right) }=0 \end{aligned}$$
(A.10)
$$\begin{aligned} \frac{h_t'(W_t)}{h_t(O_t)-h_t(V_t)}: \quad&\frac{G'\left( {\tilde{X}}_t\right) }{G\left( {\tilde{X}}_t\right) } \left[ J_t- \frac{8}{3}\frac{G'(X_t)}{G( X_t)}\frac{1}{O_t-V_t}\right] =0 \end{aligned}$$
(A.11)

Adding up the terms in (A.4A.9), we get

$$\begin{aligned} E_t \frac{h_t'(W_t)^2}{\left( h_t(V_t)-h_t(W_t)\right) \left( h_t(O_t)-h_t(W_t) \right) } \end{aligned}$$

where

$$\begin{aligned} E_t&=-2n\frac{{\tilde{X}}_t+1}{{\tilde{X}}_t}-2m\frac{{\tilde{X}}_t}{{\tilde{X}}_t +1} +\frac{4}{3} \frac{G''\left( {\tilde{X}}_t\right) }{G\left( {\tilde{X}}_t\right) } {\tilde{X}}_t ({\tilde{X}}_t +1)\\&\quad -2(l-m-n+\lambda ) +2 \frac{G'\left( {\tilde{X}}_t\right) }{G\left( {\tilde{X}}_t\right) } (2{\tilde{X}}_t+1). \end{aligned}$$

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

$$\begin{aligned} C_t&=2n \frac{X_t+1}{X_t} +2m \frac{X_t}{X_t+1} -\frac{4G''(X_t)}{3 G(X_t)} X_t (X_t+1)+2(l-m-n+\lambda )\\&\quad -2 \frac{G'(X_t)}{G(X_t)}(2X_t+1), \end{aligned}$$

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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Mathematics Subject Classification

  • Primary 60D05
  • 60J67
  • Secondary 60K35
  • 30C99