Softcover ISBN:  9782856299418 
Product Code:  AST/427 
List Price:  $68.00 
AMS Member Price:  $54.40 
Softcover ISBN:  9782856299418 
Product Code:  AST/427 
List Price:  $68.00 
AMS Member Price:  $54.40 

Book DetailsAstérisqueVolume: 427; 2021; 258 ppMSC: Primary 60; 82
There is a simple way to “glue together” a coupled pair of continuum random trees (CRTs) to produce a topological sphere. The sphere comes equipped with a measure and a spacefilling curve (which describes the “interface” between the trees). The authors present an explicit and canonical way to embed the sphere in \(\mathrm{C} \cup\{\infty\}\). In this embedding, the measure is a form of Liouville quantum gravity (LQG) with parameter \(\gamma \in (0,2)\), and the curve is spacefilling \(\mathrm{SLE}_{\kappa'}\) with \(\kappa'=16/\gamma^{2}\).
Achieving this requires the authors to develop an extensive suite of tools for working with LQG surfaces. They explain how to conformally weld socalled “quantum wedges” to obtain new quantum wedges of different weights. They construct finitevolume quantum disks and spheres of various types and give a Poissonian description of the set of quantum disks cut off by a boundaryintersecting \(\mathrm{SLE}_{\kappa'}(\rho)\) process with \(\kappa \in (0,4)\).
The authors also establish a Lévy tree description of the set of quantum disks to the left (or right) of an \(\mathrm{SLE}_{\kappa'}\) with \(\kappa' \in(4,8)\). They show that given two such trees, sampled independently, there is a.s. a canonical way to “zip them together” and recover the \(\mathrm{SLE}_{\kappa'}\).
The law of the CRT pair the authors study was shown in an earlier paper to be the scaling limit of the discrete tree/dualtree pair associated to an FKdecorated random planar map (RPM). Together, these results imply that FKdecorated RPM scales to CLEdecorated LQG in a certain “tree structure” topology.
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.
ReadershipGraduate students and research mathematicians.

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There is a simple way to “glue together” a coupled pair of continuum random trees (CRTs) to produce a topological sphere. The sphere comes equipped with a measure and a spacefilling curve (which describes the “interface” between the trees). The authors present an explicit and canonical way to embed the sphere in \(\mathrm{C} \cup\{\infty\}\). In this embedding, the measure is a form of Liouville quantum gravity (LQG) with parameter \(\gamma \in (0,2)\), and the curve is spacefilling \(\mathrm{SLE}_{\kappa'}\) with \(\kappa'=16/\gamma^{2}\).
Achieving this requires the authors to develop an extensive suite of tools for working with LQG surfaces. They explain how to conformally weld socalled “quantum wedges” to obtain new quantum wedges of different weights. They construct finitevolume quantum disks and spheres of various types and give a Poissonian description of the set of quantum disks cut off by a boundaryintersecting \(\mathrm{SLE}_{\kappa'}(\rho)\) process with \(\kappa \in (0,4)\).
The authors also establish a Lévy tree description of the set of quantum disks to the left (or right) of an \(\mathrm{SLE}_{\kappa'}\) with \(\kappa' \in(4,8)\). They show that given two such trees, sampled independently, there is a.s. a canonical way to “zip them together” and recover the \(\mathrm{SLE}_{\kappa'}\).
The law of the CRT pair the authors study was shown in an earlier paper to be the scaling limit of the discrete tree/dualtree pair associated to an FKdecorated random planar map (RPM). Together, these results imply that FKdecorated RPM scales to CLEdecorated LQG in a certain “tree structure” topology.
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.
Graduate students and research mathematicians.