Roughness and avalanches in an extended Schelling model: an explanation of urban gentrification Diego Ortega, 1 Javier Rodr´ ıguez-Laguna, 1 and Elka Korutcheva 1, 2 1 Dto. F´ ısica Fundamental, Universidad Nacional de Educaci´ on a Distancia (UNED), Spain 2 G. Nadjakov Institute of Solid State Physics, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria. (Dated: July 17, 2020) Residential segregation is analyzed via the Schelling model, in which two types of agents attempt to optimize their situation according to certain preferences and tolerance levels. Several variants of this work are focused on urban or social aspects. Whereas these models consider fixed values for wealth or tolerance, here we consider how sudden changes in the economic environment or the tolerance level affect the urban structure both in the closed city and open city frameworks, i.e. depending on whether migration processes are relevant or not. In the closed city framework, agents tend to group into clusters, whose boundary can be characterized using tools from kinetic roughening. On the other hand, in the open city approximation agents of a certain type may enter or leave the city in series of avalanches, whose statistical properties are discussed. I. INTRODUCTION People with similar features (culture, income, etc.) tend to group together in the same neighborhood, giv- ing rise to segregation on a social scale. More than 40 years ago, Schelling put forward a seminal model that de- scribes this reality, linking individual preferences to the macroscopic behaviour of the system [1]. Two different social groups, which we may call red and blue, are dis- tributed over a square lattice with some vacancies on it. Agents are characterized by a tolerance T : the fraction of different agents in their neighborhood that he or she can tolerate. The model proceeds through the following dy- namical rules: a random agent i is selected and his/her fraction of diverse neighbors is evaluated. If this frac- tion value is lower or equal than T , the agent remains at his/her location. Otherwise, he/she relocates to the nearest vacancy that meets his/her demands. For inter- mediate values of T we observe segregation, and clusters are formed with different types of agents. This model has attracted a great deal of attention, due to its simplicity and insight, giving rise to a wealth of variants. System behaviour when one kind of agents are tolerant and the vacancies are differently priced was char- acterized in [2]. Differences between constrained models, where only unhappy individuals are allowed to move, and unconstrained ones, in which all agents can relocate to vacancies as long as they keep or increase their happiness, were also studied [3]. In [4], attempted relocations suc- ceed with a probability which is modulated by a power- law linking their current happiness and the attractiveness of the offered place. The effect of the city shape, size and form is investigated in [5], finding that the properties of the system in equilibrium are weakly affected by these pa- rameters. The authors of [8] proposed a thermodynamic approach to segregation based on their cluster geometry, and considered quantities analogous to the specific heat and susceptibility, along with a connection with spin-1 models. Moreover, an open city model in which agents can leave or enter the system was described in [9]. In addition to showing different kinds of interfaces between clusters, economic aspects of the system were introduced by means of a chemical potential. Recently, the use of dif- ferent tolerance levels for the agents was proposed in [10], in a system with no vacancies, where agents could only exchange locations with agents of a different type. On the other hand, in [11] each cell of the system is consid- ered a building containing many agents, and segregation was considered both at a microscopic and a macroscopic level, giving rise to a complex phase diagram. Some of the mentioned works take into account the importance of the initial conditions [2, 10], and some others also con- sider migratory movements [9, 10]. In this article we consider how a closed city, in which no agents can enter or leave, adapts to drops in the tol- erance level. Under some circunstances, a vacancy inter- face is observed separating the main clusters. We estab- lish its statistical properties, specially its roughness. On the other hand, we also consider an open city model in which the system becomes more hostile towards one type of agent (economic handicap) and more friendly towards the other (economic advantage). This phenomenon can give rise to a partial or total overtake of the favored type of agent, which may proceed through avalanches. These avalanches are shown to present a power-law behavior which is usual of similar processes [6, 7]. Our general framework is established in similarity with the Blume- Emery-Griffiths (BEG) model in presence of an external magnetic field [8, 9]. The paper is organized as follows. In Section II we define our BEG model and discuss the dynamics of both regimes (open and closed city) and the evolution process. In Section III we describe our results for the closed city model III A and the open city model III B, linking them to well-known social mechanisms: confrontation between equal forces within the closed city or the overtaking of one kind of agents over the other due to economic supe- riority, which resembles a social process known as gen- trification [12]. Our main conclusions and proposals for further work are discussed in section IV. arXiv:2007.10767v1 [physics.soc-ph] 21 Jul 2020