scholarly journals Thick points of the Gaussian free field

2010 ◽  
Vol 38 (2) ◽  
pp. 896-926 ◽  
Author(s):  
Xiaoyu Hu ◽  
Jason Miller ◽  
Yuval Peres
Keyword(s):  
Free Field ◽  
Gaussian Free Field ◽  
Thick Points

scholarly journals The Dimension of the Boundary of a Liouville Quantum Gravity Metric Ball

2020 ◽  
Vol 378 (1) ◽  
pp. 625-689 ◽  
Author(s):  
Ewain Gwynne
Keyword(s):  
Hausdorff Dimension ◽  
Quantum Gravity ◽  
Free Field ◽  
Gaussian Free Field ◽  
Hausdorff Dimensions ◽  
Thick Points

Abstract Let $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , let h be the planar Gaussian free field, and consider the $$\gamma $$ γ -Liouville quantum gravity (LQG) metric associated with h. We show that the essential supremum of the Hausdorff dimension of the boundary of a $$\gamma $$ γ -LQG metric ball with respect to the Euclidean (resp. $$\gamma $$ γ -LQG) metric is $$2 - \frac{\gamma }{d_\gamma }\left( \frac{2}{\gamma } + \frac{\gamma }{2} \right) + \frac{\gamma ^2}{2d_\gamma ^2}$$ 2 - γ d γ 2 γ + γ 2 + γ 2 2 d γ 2 (resp. $$d_\gamma -1$$ d γ - 1 ), where $$d_\gamma $$ d γ is the Hausdorff dimension of the whole plane with respect to the $$\gamma $$ γ -LQG metric. For $$\gamma = \sqrt{8/3}$$ γ = 8 / 3 , in which case $$d_{\sqrt{8/3}}=4$$ d 8 / 3 = 4 , we get that the essential supremum of Euclidean (resp. $$\sqrt{8/3}$$ 8 / 3 -LQG) dimension of a $$\sqrt{8/3}$$ 8 / 3 -LQG ball boundary is 5/4 (resp. 3). We also compute the essential suprema of the Euclidean and $$\gamma $$ γ -LQG Hausdorff dimensions of the intersection of a $$\gamma $$ γ -LQG ball boundary with the set of metric $$\alpha $$ α -thick points of the field h for each $$\alpha \in \mathbb R$$ α ∈ R . Our results show that the set of $$\gamma /d_\gamma $$ γ / d γ -thick points on the ball boundary has full Euclidean dimension and the set of $$\gamma $$ γ -thick points on the ball boundary has full $$\gamma $$ γ -LQG dimension.

scholarly journals Thick points for a Gaussian Free Field in 4 dimensions

2015 ◽  
Vol 125 (6) ◽  
pp. 2383-2404 ◽  
Author(s):  
Alessandra Cipriani ◽  
Rajat Subhra Hazra
Keyword(s):  
Free Field ◽  
Gaussian Free Field ◽  
Thick Points

scholarly journals Crystallization of Random Matrix Orbits

2018 ◽  
Vol 2020 (3) ◽  
pp. 883-913 ◽  
Author(s):  
Vadim Gorin ◽  
Adam W Marcus
Keyword(s):  
Random Matrix ◽  
Special Functions ◽  
Free Field ◽  
Gaussian Free Field ◽  
Complex Quaternion ◽  
Cutting Out ◽  
Multiplicative Convolution ◽  
Global Asymptotics ◽  
Associated Special Functions ◽  
Derivatives Of

Abstract Three operations on eigenvalues of real/complex/quaternion (corresponding to $\beta =1,2,4$) matrices, obtained from cutting out principal corners, adding, and multiplying matrices, can be extrapolated to general values of $\beta>0$ through associated special functions. We show that the $\beta \to \infty $ limit for these operations leads to the finite free projection, additive convolution, and multiplicative convolution, respectively. The limit is the most transparent for cutting out the corners, where the joint distribution of the eigenvalues of principal corners of a uniformly-random general $\beta $ self-adjoint matrix with fixed eigenvalues is known as the $\beta $-corners process. We show that as $\beta \to \infty $ these eigenvalues crystallize on an irregular lattice consisting of the roots of derivatives of a single polynomial. In the second order, we observe a version of the discrete Gaussian Free Field put on top of this lattice, which provides a new explanation as to why the (continuous) Gaussian Free Field governs the global asymptotics of random matrix ensembles.

scholarly journals Gaussian free field in the background of correlated random clusters, formed by metallic nanoparticles

2018 ◽  
Vol 91 (5) ◽  
Author(s):  
Jafar Cheraghalizadeh ◽  
Morteza N. Najafi ◽  
Hossein Mohammadzadeh
Keyword(s):  
Metallic Nanoparticles ◽  
Free Field ◽  
Gaussian Free Field

scholarly journals Spectra of overlapping Wishart matrices and the Gaussian free field

2018 ◽  
Vol 07 (02) ◽  
pp. 1850003 ◽  
Author(s):  
Ioana Dumitriu ◽  
Elliot Paquette
Keyword(s):  
Free Field ◽  
Covariance Structure ◽  
Height Function ◽  
Gaussian Free Field ◽  
Moment Conditions ◽  
Wishart Matrices ◽  
Wigner Random Matrices ◽  
Infinite Array ◽  
Univariate Polynomials ◽  
Upper Half Plane

Consider a doubly-infinite array of i.i.d. centered variables with moment conditions, from which one can extract a finite number of rectangular, overlapping submatrices, and form the corresponding Wishart matrices. We show that under basic smoothness assumptions, centered linear eigenstatistics of such matrices converge jointly to a Gaussian vector with an interesting covariance structure. This structure, which is similar to those appearing in [A. Borodin, Clt for spectra of submatrices of Wigner random matrices, Mosc. Math. J. 14(1) (2014) 29–38; A. Borodin and V. Gorin, General beta Jacobi corners process and the Gaussian free field, preprint (2013), arXiv:1305.3627; T. Johnson and S. Pal, Cycles and eigenvalues of sequentially growing random regular graphs, Ann. Probab. 42(4) (2014) 1396–1437], can be described in terms of the height function, and leads to a connection with the Gaussian Free Field on the upper half-plane. Finally, we generalize our results from univariate polynomials to a special class of planar functions.

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