Skölunarmarkgildi slembinna auðgaðra trjáa - verkefni lokið

Fréttatilkynning verkefnisstjóra

5.7.2023

Auðguð tré eru smíðuð úr tré með því að setja endanlegt net í stað sérhvers hnúts í trénu. Unnt er að lýsa ýmsum flokkum neta á eðlilegan hátt sem auðguðum trjám og þess vegna getur skilningur á auðguðum trjám varpað ljósi á eiginleika þeirra. Einnig má nota auðguð tré til að lýsa klösum í safneðlisfræðilíkönum á slembiflötum og þau eru þannig mikilvæg viðbót við verkfærakistu stærðfræðilegrar eðlisfræði. Meðal meginniðurstaðna verkefnisins var almenn samleitniniðurstaða fyrir runu slembinna auðgaðra
trjáa sem sýndi að hún stefnir á áður óþekkt þjöppuð slembifirðrúm sem hlutu nafnið skreytt stöðug tré. Einnig voru ákveðin dæmi um auðguð tré skoðuð sérstaklega í smáatriðum. Safneðlisfræðilíkön á slembiflötum og tengsl þeirra við auðguð tré voru einnig skoðuð.

English:

The project concerns the study of random enriched trees and their scaling limits and belongs to the field of probabilty, combinatorics and mathematical physics. Enriched trees are constructed by
replacing each vertex in a tree by a finite graph. Various types of graphs may be described as
enriched trees in a natural way and therefore a good understanding of enriched trees may shed a
light on their properties. Enriched trees may also be used to describe clusters in statistical
mechanical models on random planar maps and are therefore relevant for the toolbox of
mathematical physics. The main results of the project include a general theorem for convergence of a sequence of random enriched trees towards novel random compact metric spaces which were
given the name decorated stable trees. Several explicit examples were also worked out in detail.
Statistical mechanical models on random planar maps and their connection to enriched trees were
also addressed.

∙ Information on how the results will be applied
The results of the project include development of new techniques to investigate properties of
random graphs which will be of use in the field of random planar maps and statistical mechanical
models on them. The main theorem concerning the scaling limit of random enriched trees is very
general and will have applications in the study of scaling limits of various classes of random graphs.

Á meðal niðurstaðna verkefnisins er þróun nýrrar tækni til að rannsaka eiginleika slembineta sem má hagnýta í rannsóknum á slembiflötum og safneðlisfræðilíkönum á þeim. Meginsetningin um
skölunarmarkgildi auðgaðra trjáa er mjög almenn og mun hafa hagnýtingar í rannsóknum á
skölunarmarkgildum ýmissa flokka slembineta.

∙ A list of the project's outputs
Journal Publications
Alexy, M. , Borbényi, M., Imolay, A. and Timar, A. Connectedness of the Free Uniform Spanning Forest as a function of edge weights. Electron. Commun. Probab. 27, (2022), 1 – 12. arXiv:2011.12904

Benjamini, I. and Timar, A. Invariant embeddings of unimodular random planar graphs. Electronic
Journal of Probability, 26, (2021) 1 – 18. arXiv:1910.01614

Björnberg, J., Curien, N. and Stefánsson, S. Ö. Stable shredded spheres and causal random maps with large faces. Ann. Probab. 50(5): 2056-2084 (September 2022). arXiv:1912.01378

Chen, L., Turunen, J. Ising Model on Random Triangulations of the Disk: Phase Transition. Commun. Math. Phys. (2022). https://doi.org/10.1007/s00220-022-04508-5. arXiv:2003.09343v3

Pete, G. and Timar, A. The Free Uniform Spanning Forest is disconnected in some virtually free
groups, depending on the generator set. To appear in the Annals of Probability. arXiv:2006.06387

Pete, G. and Timar, A. Finite-energy infinite clusters without anchored expansion. Bernoulli, 27,
(2021) (4) 2353 – 2361. arXiv:2011.0137

Sénizergues, D., Stefánsson, S. Ö. and Stufler, B. Decorated stable trees. Accepted for publication in Electronic Journal of Probability with minor revisions. (2022) Preprint, arXiv:2205.02968.

Stefánsson, S. Ö. and Stufler, B. Geometry of large Boltzmann outerplanar maps. Random structures and algorithms, 55(3), (October 2019), 742-771. arXiv:1710.04460

Timar, A. A factor matching of optimal tail between Poisson processes. Accepted for publication by
Combinatorica. arXiv:2106.04524

Preprints
Amankwah, D., Stefánsson, S. Ö. On scaling limits of random Halin-like maps. (2022) Preprint,
arXiv:2104.13364.

Turunen, J. Interfaces in the vertex-decorated Ising model on random triangulations of the disk.
Preprint v1-3, (2020) arXiv:2003.11012v3.

Manuscripts
Amankwah, D., Magnússon B. S. and Stefánsson, S. Ö. Decorated Brownian trees. 2022. In
preperation.

Amankwah, D. Scaling limits of random, face-weighted, tree like planar maps. PhD thesis at the
University of Iceland. In preperation.

Presentations
Amankwah, D. Scaling limits of random, face-weighted, tree like planar maps. Midway evaluation at the University of Iceland, December 11, 2020.

Stefánsson, S. Ö. Randomly decorated random trees. Mathematics seminar at the University of
Iceland, October 7, 2022.

Stefánsson, S. Ö. Scaling limits of random causal maps. 28th Nordic Congress of Mathematicians,
Aalto University, Finland, August 18-21, 2022

Stefánsson, S. Ö. Random decorated trees. Quantum geometry 2022, a conference in honour of
Bergfinnur Durhuus' 70th birthday, Faroe Islands, June 20-24, 2022.

Stefánsson, S. Ö. Random maps with large faces. Mathematics seminar at the University of Iceland,
March 31, 2022.

Stefánsson, S. Ö. Stable shredded spheres and causal random maps with large faces. Probability
seminar at Chalmers, February 15, 2022.

Stefánsson, S. Ö. Random causal maps. Random Geometry, CIRM, Marseille, France. January 20,
2022.

Stefánsson, S. Ö. Random causal maps with large faces. Probability seminar at Durham University,
January 10, 2022.

Stefánsson, S. Ö. Scaling limits of random outerplanar maps. The 10th Nordic Workshop on Statistical Physics: Biological, Complex and Non-equilibrium Systems", NORDITA, March 20 to 22, 2019.

Turunen, J. Statistical mechanics models on random lattices of the half-plane, Séminaire de physique mathématique, IPhT, Paris-Saclay, June 2022

Turunen, J. Combinatorial aspects of random planar triangulations of the disk coupled with an Ising model, Rencontres ANR COMBINÉ, Online (Lyon /Paris / Bordeaux), February 2022

Turunen, J. Ising model on random triangulations with a boundary, Rentrée de l'ANR DIMERS, Sorbonne Université, September 2021

Turunen, J. Interfaces and phase transition on random triangulations coupled with the Ising model, ALÉA 2021, CIRM, March 2021

Turunen, J. Ising model on random half-planar triangulations: critical behavior, interfaces and phase transition, Journées Cartes au CIRM, CIRM, January 2021

Turunen, J. Ising model on random triangulations of the half-plane: critical behavior and phase transition, Integrable Probability and Related Fields from a Safe Distance, Department of Mathematics, MIT, October 2020

Turunen, J. Ising model on random triangulations with a boundary: from combinatorics to geometry, Journée Cartes du 16 octobre 2020, Institut de Maths d'Orsay, Université Paris-Saclay, October 2020

Turunen, J. Ising model on random triangulations of the half-plane, Bernoulli-IMS One World Symposium 2020, August 2020 

Media coverage
Interview with Sigurður Örn Stefánsson in Morgunvaktin in Rás 1, the national radio, concerning the publication with Björnberg and Curien in the Annals of probability. August 5, 2022.