Hardcover ISBN: | 978-2-85629-952-4 |
Product Code: | COSP/28 |
List Price: | $65.00 |
AMS Member Price: | $52.00 |
Hardcover ISBN: | 978-2-85629-952-4 |
Product Code: | COSP/28 |
List Price: | $65.00 |
AMS Member Price: | $52.00 |
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Book DetailsCours SpécialisésVolume: 28; 2022; 184 ppMSC: Primary 60; 82
The Gaussian Free Field (GFF) in the continuum appears to be the natural generalisation of Brownian motion, when one replaces time by a multidimensional continuous parameter. While Brownian motion can be viewed as the most natural random real-valued function defined on \(\mathbb{R}_{+}\) with \(B(0)=0\), the GFF in a domain \(D\) of \(\mathbb{R}^{d}\) for \(d \geq 2\) is a natural random real-valued generalised function defined on \(D\) with zero boundary conditions on \(\partial D\). In particular, it is not a random continuous function.
The goal of these lecture notes is to describe some aspects of the continuum GFF and of its discrete counterpart defined on lattices, with the aim of providing a gentle self-contained introduction to some recent developments on this topic, such as the relation between the continuum GFF, Brownian loop-soups and the Conformal Loop Ensembles \(\mathrm{CLE}_{4}\).
This is an updated and expanded version of the notes written by the first author (Wendelin Werner) for graduate courses at ETH Zürich (Swiss Federal Institute of Technology in Zürich) in 2014 and 2018. It has benefited from the comments and corrections of students, as well as of a referee. The exercises that are interspersed in the first half of these notes mostly originate from the exercise sheets prepared by the second author (Ellen Powell) for this course in 2018.
A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.
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The Gaussian Free Field (GFF) in the continuum appears to be the natural generalisation of Brownian motion, when one replaces time by a multidimensional continuous parameter. While Brownian motion can be viewed as the most natural random real-valued function defined on \(\mathbb{R}_{+}\) with \(B(0)=0\), the GFF in a domain \(D\) of \(\mathbb{R}^{d}\) for \(d \geq 2\) is a natural random real-valued generalised function defined on \(D\) with zero boundary conditions on \(\partial D\). In particular, it is not a random continuous function.
The goal of these lecture notes is to describe some aspects of the continuum GFF and of its discrete counterpart defined on lattices, with the aim of providing a gentle self-contained introduction to some recent developments on this topic, such as the relation between the continuum GFF, Brownian loop-soups and the Conformal Loop Ensembles \(\mathrm{CLE}_{4}\).
This is an updated and expanded version of the notes written by the first author (Wendelin Werner) for graduate courses at ETH Zürich (Swiss Federal Institute of Technology in Zürich) in 2014 and 2018. It has benefited from the comments and corrections of students, as well as of a referee. The exercises that are interspersed in the first half of these notes mostly originate from the exercise sheets prepared by the second author (Ellen Powell) for this course in 2018.
A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.