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**Published in:** LIPIcs, Volume 225, 33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022)

We study the asymptotic behaviour of random integer partitions under a new probability law that we introduce, the Plancherel-Hurwitz measure. This distribution, which has a natural definition in terms of Young tableaux, is a deformation of the classical Plancherel measure. It appears naturally in the enumeration of Hurwitz maps, or equivalently transposition factorisations in symmetric groups.
We study a regime in which the number of factors in the underlying factorisations grows linearly with the order of the group, and the corresponding maps are of high genus. We prove that the limiting behaviour exhibits a new, twofold, phenomenon: the first part becomes very large, while the rest of the partition has the standard Vershik-Kerov-Logan-Shepp limit shape. As a consequence, we obtain asymptotic estimates for unconnected Hurwitz numbers with linear Euler characteristic, which we use to study random Hurwitz maps in this regime. This result can also be interpreted as the return probability of the transposition random walk on the symmetric group after linearly many steps.

Guillaume Chapuy, Baptiste Louf, and Harriet Walsh. Random Partitions Under the Plancherel-Hurwitz Measure, High Genus Hurwitz Numbers and Maps. In 33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 225, pp. 6:1-6:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chapuy_et_al:LIPIcs.AofA.2022.6, author = {Chapuy, Guillaume and Louf, Baptiste and Walsh, Harriet}, title = {{Random Partitions Under the Plancherel-Hurwitz Measure, High Genus Hurwitz Numbers and Maps}}, booktitle = {33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022)}, pages = {6:1--6:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-230-3}, ISSN = {1868-8969}, year = {2022}, volume = {225}, editor = {Ward, Mark Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2022.6}, URN = {urn:nbn:de:0030-drops-160921}, doi = {10.4230/LIPIcs.AofA.2022.6}, annote = {Keywords: Random partitions, limit shapes, transposition factorisations, map enumeration, Hurwitz numbers, RSK algorithm, giant components} }

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**Published in:** LIPIcs, Volume 60, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)

A class of graphs is bridge-addable if given a graph G in the class, any graph obtained by adding an edge between two connected components of G is also in the class. The authors recently proved a conjecture of McDiarmid, Steger, and Welsh stating that if G is bridge-addable and G_n is a uniform n-vertex graph from G, then G_n is connected with probability at least (1+o(1))e^{-1/2}. The constant e^{-1/2} is best possible since it is reached for the class of forests.
In this paper we prove a form of uniqueness in this statement: if G is a bridge-addable class and the random graph G_n is connected with probability close to e^{-1/2}, then G_n is asymptotically close to a uniform forest in some "local" sense. For example, if the probability converges to e^{-1/2}, then G_n converges for the Benjamini-Schramm topology, to the uniform infinite random forest F_infinity. This result is reminiscent of so-called "stability results" in extremal graph theory, with the difference that here the "stable" extremum is not a graph but a graph class.

Guillaume Chapuy and Guillem Perarnau. Local Convergence and Stability of Tight Bridge-Addable Graph Classes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 26:1-26:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chapuy_et_al:LIPIcs.APPROX-RANDOM.2016.26, author = {Chapuy, Guillaume and Perarnau, Guillem}, title = {{Local Convergence and Stability of Tight Bridge-Addable Graph Classes}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)}, pages = {26:1--26:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-018-7}, ISSN = {1868-8969}, year = {2016}, volume = {60}, editor = {Jansen, Klaus and Mathieu, Claire and Rolim, Jos\'{e} D. P. and Umans, Chris}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2016.26}, URN = {urn:nbn:de:0030-drops-66494}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2016.26}, annote = {Keywords: bridge-addable classes, random graphs, stability, local convergence, random forests} }