# From Spin Glasses to Branching Brownian Motion—and Back?

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## Abstract

I review recent work on the construction of the extremal process of branching Brownian motion. I place this in the context of spin glass theory and in particular the Generalised Random Energy models of Derrida and Gardner. The main emphasis is on a review of the construction of the extremal process of branching Brownian motion, done in collaboration with Louis-Pierre Arguin and Nicola Kistler (Commun Pure Appl Math 64(12):1647–1676, 2011; Ann Appl Probab 22(4):1693–1711, 2012; Probab Theory Relat Fields 157:535–574, 2013). I also present some more recent results on the variable speed Brownian motion, obtained with Lisa Hartung (Electron J Probab 19(18):1–28, 2014; Lat Am J Probab Math Stat 12:261–291, 2015).

## Keywords

Brownian Motion Point Process Gaussian Process Gibbs Measure Brownian Bridge## Notes

### Acknowledgements

I am deeply grateful to my collaborators on the matters of these lectures, Louis-Pierre Arguin, Nicola Kistler, Irina Kurkova, and, most recently, Lisa Hartung. They did the bulk of the work, and without them none this would have been achieved. I also thank Jiří Černý, Lisa Hartung, and an anonymous referee for pointing out various mistakes and suggesting improvement. I thank Marek Biskup, Jiří Černý, and Roman Kotecký for organising the excellent school in Prague and for the opportunity to present these lectures there.

This work is partially supported through the German Research Foundation in the Collaborative Research Center 1060 “The Mathematics of Emergent Effects”, the Priority Programme 1590 “Probabilistic Structures in Evolution”, the Hausdorff Center for Mathematics (HCM), and the Cluster of Excellence “ImmunoSensation” at Bonn University.

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