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Self-Avoiding Walks on Random Quadrangulations

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The local limit of random quadrangulations (UIPQ) and the local limit of quadrangulations with a simple boundary (the simple boundary UIHPQ) are two very well studied objects. We shall see how the simple boundary UIHPQ relates to an annealed model of self-avoiding walk on random quadrangulations, and how metric information obtained for the UIHPQ can be used to study quantities such as the displacement of the self-avoiding walk from the origin, as well as to ultimately investigate how the biasing of random quadrangulations by the number of their self-avoiding walks affects their local limit.

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Croisements inéluctables et preuves au hasard

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14212568_1199046323500737_1042786532754096278_nIf you're ever in Lyon on a Wednesday, make sure not to miss the Séminaire de la détente at the MMI! Cake, tidbits of mathematical fun, plus even a great apéro afterwords: what's there not to love?!
Was so happy to give a talk there in September: thanks to Olga and Marie for the invitation and delicious cake! Here's the abstract and slides (as usual, a version in a better format, possibly with some added explanations – some essential bits were done on the blackboard – may or may not be on the way):

For an English version of the following abstract, click here.

Nous sommes en 1944 ; le mathématicien Pál Turán, qui se trouve dans un camp de travail près de Budapest, est chargé de transporter des briques des fours aux aires de stockage via des chariots sur rails. Chaque four est relié par des rails à chacune des aires de stockage ; pousser les chariots ne demande pas trop d’effort, sauf aux croisements de deux rails, où ils ont tendance à capoter, de sorte que la plupart des briques tombent dehors. Pourquoi – s’interroge Pál – construire ce réseau de manière aussi inefficace, avec tellement de croisements exaspérants ?! Comment pourrait-on minimiser le nombre de croisements, tout en reliant chaque four à chaque aire de stockage ?
C’est peut-être la première question (qui est encore essentiellement ouverte !) portant sur le ’nombre de croisements’, une notion importante en théorie des graphes. En l’étudiant, on va ’croiser’ des artistes constructivistes, des informaticiens, des matheux sceptiques, et on va finalement tomber sur l’une des parties les plus précieuses de l’héritage de Pál Erdős : la méthode que l’on appelle ’probabiliste’.

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The scaling limit of random outerplanar maps

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Third take on a talk about outerplanar maps, and perhaps the most complete account I ever gave of this paper; the slides are the same those of the Journée YSP, with an extra section that describes the core algorithm. Sorry about the annoying video format; an incredibly heavy pdf file is available here.

A planar map is outerplanar if all of its vertices belong to a single (outer)face. The scaling limit for various classes of large planar maps has been shown to be the so-called “Brownian Map”; under certain conditions, however, different asymptotic behaviours may emerge, and some classes of planar maps with a macroscopic face admit Aldous’ CRT, or a scalar multiple thereof, as the scaling limit. We shall see that this is the case for outerplanar maps as well: using a bijection by Bonichon, Gavoille and Hanusse one can show that uniform outerplanar maps with n vertices, suitably rescaled by a factor 1/\sqrt{n}, converge to 7\sqrt{2}/9 times the CRT.

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The Scaling Limit of Random Outerplanar Maps

Video

A planar map is outerplanar if all of its vertices belong to a single (outer)face. The scaling limit for various classes of large planar maps has been shown to be the so-called “Brownian Map”; under certain conditions, however, different asymptotic behaviours may emerge, and some classes of planar maps with a macroscopic face admit Aldous’ CRT, or a scalar multiple thereof, as the scaling limit. We shall see that this is the case for outerplanar maps as well: using a bijection by Bonichon, Gavoille and Hanusse one can show that uniform outerplanar maps with n vertices, suitably rescaled by a factor 1/\sqrt{n}, converge to 7\sqrt{2}/9 times the CRT.