scholarly journals Return Probability and Recurrence for the Random Walk Driven by Two-Dimensional Gaussian Free Field

2019 ◽  
Vol 373 (1) ◽  
pp. 45-106 ◽  
Author(s):  
Marek Biskup ◽  
Jian Ding ◽  
Subhajit Goswami
Keyword(s):  
Random Walk ◽  
Free Field ◽  
Two Dimensional ◽  
Gaussian Free Field ◽  
Return Probability

scholarly journals Quantum walks and elliptic integrals

2010 ◽  
Vol 20 (6) ◽  
pp. 1091-1098 ◽  
Author(s):  
NORIO KONNO
Keyword(s):  
Random Walk ◽  
Generating Function ◽  
Elliptic Integral ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Elliptic Integrals ◽  
Two Dimensional ◽  
One Dimensional ◽  
Similar Expression ◽  
Return Probability

Pólya showed in his 1921 paper that the generating function of the return probability for a two-dimensional random walk can be written in terms of an elliptic integral. In this paper we present a similar expression for a one-dimensional quantum walk.

scholarly journals Contour lines of the two-dimensional discrete Gaussian free field

2009 ◽  
Vol 202 (1) ◽  
pp. 21-137 ◽  
Author(s):  
Oded Schramm ◽  
Scott Sheffield
Keyword(s):  
Free Field ◽  
Two Dimensional ◽  
Gaussian Free Field ◽  
Contour Lines

scholarly journals Extreme values for two-dimensional discrete Gaussian free field

2014 ◽  
Vol 42 (4) ◽  
pp. 1480-1515 ◽  
Author(s):  
Jian Ding ◽  
Ofer Zeitouni
Keyword(s):  
Extreme Values ◽  
Free Field ◽  
Two Dimensional ◽  
Gaussian Free Field

scholarly journals ON BOUNDED-TYPE THIN LOCAL SETS OF THE TWO-DIMENSIONAL GAUSSIAN FREE FIELD

2017 ◽  
Vol 18 (3) ◽  
pp. 591-618 ◽  
Author(s):  
Juhan Aru ◽  
Avelio Sepúlveda ◽  
Wendelin Werner
Keyword(s):  
Harmonic Function ◽  
Connected Domain ◽  
Free Field ◽  
Mean Value ◽  
Two Dimensional ◽  
Gaussian Free Field ◽  
Simply Connected Domain ◽  
The Mean ◽  
Conformal Loop Ensemble ◽  
Simply Connected

We study certain classes of local sets of the two-dimensional Gaussian free field (GFF) in a simply connected domain, and their relation to the conformal loop ensemble$\text{CLE}_{4}$and its variants. More specifically, we consider bounded-type thin local sets (BTLS), where thin means that the local set is small in size, and bounded type means that the harmonic function describing the mean value of the field away from the local set is bounded by some deterministic constant. We show that a local set is a BTLS if and only if it is contained in some nested version of the$\text{CLE}_{4}$carpet, and prove that all BTLS are necessarily connected to the boundary of the domain. We also construct all possible BTLS for which the corresponding harmonic function takes only two prescribed values and show that all these sets (and this includes the case of$\text{CLE}_{4}$) are in fact measurable functions of the GFF.

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