# Self-organized Segregation on the Grid

- 118 Downloads
- 1 Citations

## Abstract

We consider an agent-based model with exponentially distributed waiting times in which two types of agents interact locally over a graph, and based on this interaction and on the value of a common intolerance threshold \(\tau \), decide whether to change their types. This is equivalent to a zero-temperature ising model with Glauber dynamics, an asynchronous cellular automaton with extended Moore neighborhoods, or a Schelling model of self-organized segregation in an open system, and has applications in the analysis of social and biological networks, and spin glasses systems. Some rigorous results were recently obtained in the theoretical computer science literature, and this work provides several extensions. We enlarge the intolerance interval leading to the expected formation of large segregated regions of agents of a single type from the known size \(\epsilon >0\) to size \(\approx 0.134\). Namely, we show that for \(0.433< \tau < 1/2\) (and by symmetry \(1/2<\tau <0.567\)), the expected size of the largest segregated region containing an arbitrary agent is exponential in the size of the neighborhood. We further extend the interval leading to expected large segregated regions to size \(\approx 0.312\) considering “almost segregated” regions, namely regions where the ratio of the number of agents of one type and the number of agents of the other type vanishes quickly as the size of the neighborhood grows. In this case, we show that for \(0.344 < \tau \le 0.433\) (and by symmetry for \(0.567 \le \tau <0.656\)) the expected size of the largest almost segregated region containing an arbitrary agent is exponential in the size of the neighborhood. This behavior is reminiscent of supercritical percolation, where small clusters of empty sites can be observed within any sufficiently large region of the occupied percolation cluster. The exponential bounds that we provide also imply that complete segregation, where agents of a single type cover the whole grid, does not occur with high probability for \(p=1/2\) and the range of intolerance considered.

## Keywords

Unperturbed Schelling segregation Zero-temperature ising model Asynchronous cellular automation (ACA) Percolation theory First passage percolation Exponential segregation## Notes

### Acknowledgements

The authors thank Prof. Jason Schweinsberg of the Mathematics Department of University of California at San Diego for providing invaluable feedback on earlier drafts of the paper and for suggesting some improved proofs.

## References

- 1.Arratia, R.: Site recurrence for annihilating random walks on \({\mathbb{Z}}^d\). Ann. Probab.
**11**, 706–713 (1983)MathSciNetCrossRefMATHGoogle Scholar - 2.Barmpalias, G., Elwes, R., Lewis-Pye, A.: Digital morphogenesis via Schelling segregation. In: Foundations of Computer Science (FOCS), 2014 IEEE 55th Annual Symposium on, pp. 156–165. IEEE (2014)Google Scholar
- 3.Barmpalias, G., Elwes, R., Lewis-Pye, A.: Minority population in the one-dimensional Schelling model of segregation (2015). arXiv:1508.02497
- 4.Barmpalias, G., Elwes, R., Lewis-Pye, A.: Tipping points in 1-dimensional Schelling models with switching agents. J. Stat. Phys.
**158**(4), 806–852 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar - 5.Barmpalias, G., Elwes, R., Lewis-Pye, A.: Unperturbed Schelling segregation in two or three dimensions. J. Stat. Phys.
**164**(6), 1460–1487 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar - 6.Bhakta, P., Miracle, S., Randall, D.: Clustering and mixing times for segregation models on \({\mathbb{Z}} ^d\). In: Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 327–340. Society for Industrial and Applied Mathematics (2014)Google Scholar
- 7.Brandt, C., Immorlica, N., Kamath, G., Kleinberg, R.: An analysis of one-dimensional Schelling segregation. In: Proceedings of the Forty-Fourth Annual ACM Symposium on Theory of Computing, pp. 789–804. ACM (2012)Google Scholar
- 8.Caputo, P., Martinelli, F.: Phase ordering after a deep quench: the stochastic ising and hard core gas models on a tree. Probab. Theory Relat Fields
**136**(1), 37–80 (2006)MathSciNetCrossRefMATHGoogle Scholar - 9.Castellano, C., Fortunato, S., Loreto, V.: Statistical physics of social dynamics. Rev. Modern Phys.
**81**(2), 591–646 (1969)ADSCrossRefGoogle Scholar - 10.Chopard, B., Droz, M.: Cellular Automata. Springer, Berlin (1998)MATHGoogle Scholar
- 11.Damron, M., Hanson, J., Sosoe, P., et al.: Subdiffusive concentration in first-passage percolation. Electron. J. Probab.
**19**(109), 1–27 (2014)MathSciNetMATHGoogle Scholar - 12.Draief, M., Massouli, L.: Epidemics and Rumours in Complex Networks. Cambridge University Press, Cambridge (2010)MATHGoogle Scholar
- 13.Easley, D., Kleinberg, J.: Networks, Crowds, and Markets: Reasoning About a Highly Connected World. Cambridge University Press, Cambridge (2010)CrossRefMATHGoogle Scholar
- 14.Erdos, P., Ney, P.: Some problems on random intervals and annihilating particles. Ann. Probab.
**2**(5), 828–839 (1974)MathSciNetCrossRefMATHGoogle Scholar - 15.Fontes, L.R., Schonmann, R., Sidoravicius, V.: Stretched exponential fixation in stochastic ising models at zero temperature. Commun. Math. Phys.
**228**(3), 495–518 (2002)ADSMathSciNetCrossRefMATHGoogle Scholar - 16.Fortuin, C.M., Kasteleyn, P.W., Ginibre, J.: Correlation inequalities on some partially ordered sets. Commun. Math. Phys.
**22**(2), 89–103 (1971)ADSMathSciNetCrossRefMATHGoogle Scholar - 17.Garet, O., Marchand, R.: Large deviations for the chemical distance in supercritical Bernoulli percolation. Ann. Probab.
**35**(3), 833–866 (2007)MathSciNetCrossRefMATHGoogle Scholar - 18.Grimmett, G.: Percolation, vol. 321, 2nd edn. Springer, Berlin (1999)CrossRefMATHGoogle Scholar
- 19.Harris, T.E.: A correlation inequality for Markov processes in partially ordered state spaces. Ann. Probab.
**5**(3), 451–454 (1977)MathSciNetCrossRefMATHGoogle Scholar - 20.Hethcote, H.W.: The mathematics of infectious diseases. SIAM Rev.
**42**(4), 599–653 (2000)ADSMathSciNetCrossRefMATHGoogle Scholar - 21.Immorlica, N., Kleinberg, R., Lucier, B., Zadomighaddam, M.: Exponential segregation in a two-dimensional schelling model with tolerant individuals. In: Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 984–993. SIAM (2017)Google Scholar
- 22.Jackson, M.O., Watts, A.: On the formation of interaction networks in social coordination games. Games Economic Behav.
**41**(2), 265–291 (2002)MathSciNetCrossRefMATHGoogle Scholar - 23.Janson, S., Luczak, T., Rucinski, A.: Random Graphs, vol. 45. Wiley, London (2011)MATHGoogle Scholar
- 24.Kanoria, Y., Montanari, A., et al.: Majority dynamics on trees and the dynamic cavity method. Ann. Appl. Probab.
**21**(5), 1694–1748 (2011)MathSciNetCrossRefMATHGoogle Scholar - 25.Kesten, H.: On the speed of convergence in first-passage percolation. Ann. Appl. Probab.
**3**(2), 296–338 (1993)MathSciNetCrossRefMATHGoogle Scholar - 26.Kleinberg, J.: Cascading behavior in networks: algorithmic and economic issues. Algorithmic Game Theory
**24**, 613–632 (2007)MathSciNetCrossRefMATHGoogle Scholar - 27.Liggett, T.: Interacting Particle Systems, vol. 276. Springer Science & Business Media, New York (2012)MATHGoogle Scholar
- 28.Liggett, T.M.: Stochastic models for large interacting systems and related correlation inequalities. Proc. Natl. Acad. Sci.
**107**(38), 16413–16419 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar - 29.Liggett, T.M.: Stochastic interacting systems: contact, voter and exclusion processes, vol. 324. Springer Science & Business Media (2013)Google Scholar
- 30.Meyer-Ortmanns, H.: Immigration, integration and ghetto formation. Int. J. Modern Phys. C
**14**(03), 311–320 (2003)ADSMathSciNetCrossRefGoogle Scholar - 31.Mobius, M.M., Rosenblat, T.: The formation of ghettos as a local interaction phenomenon. Unpublished manuscript, Harvard University (2000)Google Scholar
- 32.Morris, R.: Zero-temperature glauber dynamics on \(\mathbb{Z}^d\). Probab. Theory Relat. Fields
**149**(3–4), 417–434 (2011)MathSciNetCrossRefMATHGoogle Scholar - 33.Schelling, T.C.: Models of segregation. Am. Econ. Rev.
**59**(2), 488–493 (1969)Google Scholar - 34.Schelling, T.C.: Dynamic models of segregation. J. Math. Sociol.
**1**(2), 143–186 (1971)CrossRefMATHGoogle Scholar - 35.Schulze, C.: Potts-like model for ghetto formation in multi-cultural societies. Int. J. Modern Phys. C
**16**(03), 351–355 (2005)ADSCrossRefGoogle Scholar - 36.Stauffer, D., Solomon, S.: Ising, Schelling and self-organising segregation. Eur. Phys. J. B
**57**(4), 473–479 (2007)ADSCrossRefGoogle Scholar - 37.Talagrand, M.: Concentration of measure and isoperimetric inequalities in product spaces. Publications Mathematiques de l’IHES
**81**(1), 73–205 (1995)MathSciNetCrossRefMATHGoogle Scholar - 38.Young, H.P.: Individual Strategy and Social Structure: An Evolutionary Theory of Institutions. Princeton University Press, Princeton (2001)Google Scholar
- 39.Zhang, J.: A dynamic model of residential segregation. J. Math. Sociol.
**28**(3), 147–170 (2004)CrossRefMATHGoogle Scholar - 40.Zhang, J.: Residential segregation in an all-integrationist world. J. Econ. Behav. Organ.
**54**(4), 533–550 (2004)CrossRefGoogle Scholar - 41.Zhang, J.: Tipping and residential segregation: a unified Schelling model. J. Reg. Sci.
**51**(1), 167–193 (2011)ADSCrossRefGoogle Scholar