## Research interests

My background is in model theory, part of mathematical logic with applications to algebra, but I have broad interests. Recently, I have become interested in the evolution of random processes, and in networks built by such processes. These are structures traditionally used by "applied" mathematicians to model real-world phenomena (such as the emergence of racial segregation, or the growth of the internet). I am interested in what we might be able to say about such structures using the "pure" mathematical language of combinatorics and logic.

### Useful links

My blog and personal webpage

## Publications

**Barmpalias G, Elwes R, Lewis-Pye A** Unperturbed Schelling Segregation in Two or Three Dimensions *Journal of Statistical Physics*, **164**, 1460-1487, 2016

DOI:10.1007/s10955-016-1589-6

View abstract

*Schelling’s models of segregation, first described in 1969 (Am Econ Rev 59:488–493, 1969) are among the best known models of self-organising behaviour. Their original purpose was to identify mechanisms of urban racial segregation. But his models form part of a family which arises in statistical mechanics, neural networks, social science, and beyond, where populations of agents interact on networks. Despite extensive study, unperturbed Schelling models have largely resisted rigorous analysis, prior results generally focusing on variants in which noise is introduced into the dynamics, the resulting system being amenable to standard techniques from statistical mechanics or stochastic evolutionary game theory (Young in Individual strategy and social structure: an evolutionary theory of institutions, Princeton University Press, Princeton, 1998). A series of recent papers (Brandt et al. in: Proceedings of the 44th annual ACM symposium on theory of computing (STOC 2012), 2012); Barmpalias et al. in: 55th annual IEEE symposium on foundations of computer science, Philadelphia, 2014, J Stat Phys 158:806–852, 2015), has seen the first rigorous analyses of 1-dimensional unperturbed Schelling models, in an asymptotic framework largely unknown in statistical mechanics. Here we provide the first such analysis of 2- and 3-dimensional unperturbed models, establishing most of the phase diagram, and answering a challenge from Brandt et al. in: Proceedings of the 44th annual ACM symposium on theoryof computing (STOC 2012), 2012).*

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**Barmpalias G, Elwes R, Lewis-Pye A** Tipping Points in 1-dimensional Schelling Models with Switching Agents *Journal of Statistical Physics*, **158**, 806-852, 2015

DOI:10.1007/s10955-014-1141-5

View abstract

*Schelling’s spacial proximity model was an early agent-based model, illustrating how ethnic segregation can emerge, unwanted, from the actions of citizens acting according to individual local preferences. Here a 1-dimensional unperturbed variant is studied under switching agent dynamics, interpretable as being open in that agents may enter and exit the model. Following the authors’ work (Barmpalias et al., FOCS, 2014) and that of Brandt et al. (Proceedings of the 44th ACM Symposium on Theory of Computing (STOC 2012), 2012), rigorous asymptotic results are established. The dynamic allows either typeto take over almost everywhere. Tipping points are identified between the regions of takeover and staticity. In a generalization of the models considered in [1] and [3], the model’s parameters comprise the initial proportions of the two types, along with independent values of the tolerance for each type. This model comprises a 1-dimensional spin-1 model with spin dependent external field, as well as providing an example of cascading behaviour within a network.*

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**Barmpalias G; Elwes RH; Lewis-Pye A ** *Digital Morphogenesis via Schelling Segregation*IEEE Annual Symposium on Foundations of Computer Science, 156-165 2014

DOI:10.1109/FOCS.2014.25

View abstract

*Schelling's model of segregation looks to explain the way in which particles or agents of two types may come to arrange themselves spatially into configurations consisting of large homogeneous clusters, i.e. connected regions consisting of only one type. As one of the earliest agent based models studied by economists and perhaps the most famous model of self-organising behaviour, it also has direct links to areas at theinterface between computer science and statistical mechanics, such as the Ising model and the study of contagion and cascading phenomena in networks. While the model has been extensively studied it has largely resisted rigorous analysis, prior results from the literature generally pertaining to variants of the model which are tweaked so as to be amenable to standard techniques from statistical mechanics or stochastic evolutionary game theory. In BK, Brandt, Immorlica, Kamath and Kleinberg provided the first rigorous analysis of the unperturbed model, for a specific set of input parameters. Here we provide a rigorous analysis of the model's behaviour much more generally and establish some surprising forms of threshold behaviour, notably the existence of situations where an increased level of intolerance for neighbouring agents of opposite type leads almost certainly to decreased segregation.*

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**Elwes RH, Jaligot, Macpherson HD** Groups in supersimple and pseudofinite theories *Proceedings of the London Mathematical Society*, **103**, 1049-1082, 2011

DOI:10.1112/plms/pdr002