Journal article

Proof of the WARM whisker conjecture for neuronal connections

Mark Holmes, Victor Kleptsyn

Chaos | AIP Publishing | Published : 2017

Abstract

This paper is devoted to the study of the so-called WARM reinforcement models that are generalisations of Pólya's urn. We show that in the graph setting, once the exponent α of the reinforcement function is greater than 2, the stable and critical equilibria can be supported only on spanning forests, and once α > 25, on spanning whisker forests. Thus, we prove the whisker forests conjecture from Hofstad et al. [Ann. Appl. Probab. 26(4), 2494–2539 (2016)].

University of Melbourne Researchers

Grants

Awarded by Australian Research Council's Discovery Programme


Awarded by RFBR project


Funding Acknowledgements

The authors are very grateful to the organizers of the conference "Spin glasses, random graphs and percolation" for bringing them together, and for Institute Henri Poincare for its kind hospitality. We thank an anonymous referee for helpful comments. This research was supported under Australian Research Council's Discovery Programme (Future Fellowship project No. FT160100166). V.K. also thanks the support of RFBR project No. 16-01-00748-a.