# zbMATH — the first resource for mathematics

Thick points of the Gaussian free field. (English) Zbl 1201.60047
Let $$U\subseteq \mathbb{C}$$ be a bounded domain with smooth boundary and let $$F$$ be an instance of the continuum Gaussian free field on $$U$$ with respect to the Dirichlet inner product $$\int_{U}\nabla f\left( x\right) \cdot \nabla g\left( x\right) \mathrm{d}x$$. The set $$T\left( a;U\right)$$ of $$a$$-thick points of $$F$$ consists of those $$z\in U$$ such that the average of $$F$$ on a disk of radius $$r$$ centered at $$z$$ has growth $$\sqrt{a/\pi }\log \displaystyle\frac{1}{r}$$ as$$\;r\rightarrow 0$$. The authors show that for each $$0\leq a\leq 2$$ the Hausdorff dimension of $$T\left( a;U\right)$$ is almost surely $$2-a,$$ that $$v_{2-a}\left( T\left( a;U\right) \right) =\infty$$ when $$0<a\leq 2$$ and $$\nu _{2}\left( T\left( 0;U\right) \right) =\nu _{2}\left( U\right)$$ almost surely, where $$\nu _{\alpha }$$ is a Hausdorff-$$\alpha$$ measure, and that $$T\left( a;U\right)$$ is almost surely empty when $$a>2$$. Furthermore, they prove that $$T\left( a;U\right)$$ is invariant under conformal transformations in an appropriate sense. The notion of a thick point is connected to the Liouville quantum gravity measure with parameter $$\gamma$$ given formally by $$\Gamma \left( \mathrm{d}z\right) =e^{\sqrt{2\pi }\gamma F\left( z\right) }\mathrm{d}z$$ considered by B. Duplantier and S. Sheffield [arXiv:0808.1560].

##### MSC:
 60G60 Random fields 60G15 Gaussian processes 60G18 Self-similar stochastic processes 28A80 Fractals
Full Text:
##### References:
 [1] Ben Arous, G. and Deuschel, J.-D. (1996). The construction of the ( d +1)-dimensional Gaussian droplet. Comm. Math. Phys. 179 467-488. · Zbl 0858.60096 · doi:10.1007/BF02102597 [2] Bolthausen, E., Deuschel, J.-D. and Giacomin, G. (2001). Entropic repulsion and the maximum of the two-dimensional harmonic crystal. Ann. Probab. 29 1670-1692. · Zbl 1034.82018 · doi:10.1214/aop/1015345767 [3] Daviaud, O. (2005). Thick points for the Cauchy process. Ann. Inst. H. Poincaré Probab. Statist. 41 953-970. · Zbl 1074.60084 · doi:10.1016/j.anihpb.2004.10.001 · numdam:AIHPB_2005__41_5_953_0 · eudml:77876 [4] Daviaud, O. (2006). Extremes of the discrete two-dimensional Gaussian free field. Ann. Probab. 34 962-986. · Zbl 1104.60062 · doi:10.1214/009117906000000061 [5] Dembo, A., Peres, Y., Rosen, J. and Zeitouni, O. (1999). Thick points for transient symmetric stable processes. Electron. J. Probab. 4 13. · Zbl 0927.60077 · emis:journals/EJP-ECP/EjpVol4/paper9.abs.html · eudml:120048 [6] Dembo, A., Peres, Y., Rosen, J. and Zeitouni, O. (2000). Thick points for spatial Brownian motion: Multifractal analysis of occupation measure. Ann. Probab. 28 1-35. · Zbl 1130.60311 · doi:10.1214/aop/1019160110 · euclid:aop/1019160110 [7] Dembo, A., Peres, Y., Rosen, J. and Zeitouni, O. (2001). Thick points for planar Brownian motion and the Erdős-Taylor conjecture on random walk. Acta Math. 186 239-270. · Zbl 1008.60063 · doi:10.1007/BF02401841 · www.actamathematica.org [8] Deuschel, J.-D. and Giacomin, G. (2000). Entropic repulsion for massless fields. Stochastic Process. Appl. 89 333-354. · Zbl 1045.60103 · doi:10.1016/S0304-4149(00)00030-2 [9] Duplantier, B. and Sheffield, S. (2009). Liouville quantum gravity and KPZ. Available at · Zbl 1226.81241 · doi:10.1103/PhysRevLett.102.150603 [10] Evans, L. (2002). Partial Differential Equations. American Mathematical Society Translations 206 . Amer. Math. Soc., Providence, RI. · Zbl 1042.65027 [11] Janson, S. (1997). Gaussian Hilbert Spaces. Cambridge Tracts in Mathematics 129 . Cambridge Univ. Press, Cambridge. · Zbl 0887.60009 [12] Karatzas, I. and Shreve, S. E. (1988). Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics 113 . Springer, New York. · Zbl 0638.60065 [13] Katznelson, Y. (2004). An Introduction to Harmonic Analysis , 3rd ed. Cambridge Univ. Press, Cambridge. · Zbl 1055.43001 [14] Kenyon, R. (2001). Dominos and the Gaussian free field. Ann. Probab. 29 1128-1137. · Zbl 1034.82021 · doi:10.1214/aop/1015345599 [15] Naddaf, A. and Spencer, T. (1997). On homogenization and scaling limit of some gradient perturbations of a massless free field. Comm. Math. Phys. 183 55-84. · Zbl 0871.35010 · doi:10.1007/BF02509796 [16] Orey, S. and Pruitt, W. E. (1973). Sample functions of the N -parameter Wiener process. Ann. Probab. 1 138-163. · Zbl 0284.60036 · doi:10.1214/aop/1176997030 [17] Randol, B. (1969). On the Fourier transform of the indicator function of a planar set. Trans. Amer. Math. Soc. 139 271-278. · Zbl 0183.26904 · doi:10.2307/1995319 [18] Revuz, D. and Yor, M. (2004). Continuous Martingales and Brownian Motion . Springer, Berlin. · Zbl 1087.60040 [19] Rider, B. and Virag, B. (2007). The noise in the circular law and the Gaussian free field. Int. Math. Res. Not. IMRN 2007 Art. ID rnm006. · Zbl 1130.60030 · doi:10.1093/imrn/rnm006 [20] Sheffield, S. (2007). Gaussian free fields for mathematicians. Probab. Theory Related Fields 139 521-541. · Zbl 1132.60072 · doi:10.1007/s00440-006-0050-1 [21] Taylor, M. E. (1996). Partial Differential Equations. I. Basic Theory. Applied Mathematical Sciences 115 . Springer, New York. · Zbl 0869.35002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.