Emergence of Condensation in Kingman’s Model of Selection and Mutation
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We describe the onset of condensation in the simple model for the balance between selection and mutation given by Kingman in terms of a scaling limit theorem. Loosely speaking, this shows that the wave moving towards genes of maximal fitness has the shape of a gamma distribution. We conjecture that this wave shape is a universal phenomenon that can also be found in a variety of more complex models, well beyond the genetics context, and provide some further evidence for this.
KeywordsHouse of cards model Bose-Einstein condensation Condensation wave Selection Mutation Gamma distribution Universality Random permutations Random network Fitness distribution
Mathematics Subject Classification92D10 37N25 82B26 92D15
P. Mörters acknowledges useful discussions with Daniel Ueltschi at the Oberwolfach workshop Interplay of analysis and probability in physics, January 2012. We would like to thank Marcel Ortgiese for agreeing to include a preview of  in our discussion.
- 5.Dereich, S.: Preferential attachment with fitness: unfolding Bose-Einstein condensation. (2012, in preparation) Google Scholar
- 6.Dereich, S., Ortgiese, M.: Robust analysis of preferential attachment models with fitness. (2012, in preparation) Google Scholar
- 7.Ercolani, N.M., Ueltschi, D.: Cycle structure of random permutations with cycle weights. Random Struct. Algorithms (2012, to appear). doi: 10.1002/rsa.20430
- 8.Feller, W.: An Introduction to Probability Theory and Its Applications. Volume I, 3rd edn. Wiley, New York (1968) Google Scholar