Coarsening process in the 2d voter model

Alessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 1 / 34 Coarsening process in the 2d voter model Alessandro Tartaglia LPTHE, Université Pierre et Marie Curie alessandro.tartaglia91@gmail.com May 8, 2015 Supervisors: Leticia F. Cugliandolo, Marco Picco

Alessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 2 / 34 Overview 1 Definition and general properties of the model Introduction and definition of the model Analytical results Duality 2 Statistics of domain areas and coarsening Snapshots Time evolution of the domain area distribution Percolating regime 3 Conlusions

Alessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 3 / 34 Introduction The voter model is one of the many stochastic models that are used to describe problems relative to opinion dynamics. It was first introduced by Clifford and Sudbury Clifford-Sudbury [?] as a model for the competition of species and it is still widely used as a simple model to characterize the role of space in ecology. Applications of the model can be found also in condensed matter physics, such as for examples in the study of the dimer-dimer heterogeneous catalysis in the reaction-controlled limit [?], Evans-Ray in the study of spinodal decomposition in fluids [?] Scheucher-Spohn and in general in the context of coarsening processes in absence of surface tension.

Alessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 4 / 34 General definition of the model The general definition of the model is extremely simple. Each node i of a connected graph is endowed with a binary variable σ i, say for example σ i { 1, 1}. The opinions of any given voter changes at random times under the influence of opinions of his neighbours. At random times, a random individual is selected and that voter s opinion are changed according to a stochastic rule. An alternative interpretation for the model is in terms of spatial conflict: suppose two nations control the areas (sets of nodes) labelled 0 or 1. A flip from 0 to 1 at a given location indicates an invasion of that site by the other nation and thus sometimes the model is referred also as invasion process.

Alessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 5 / 34 Definition of the model Usually the model is defined on a regular lattice, in particular, the d-dimensional integer lattice Z d. W(x, σ) denotes the rate at which the site x Z d changes opinion, given that the system is in the configuration σ { 1, 1} Zd In the most simple cases W(x, σ) is taken isotropic and short-ranged. For the basic linear voter model, the spin-flip rate is just proportional to the number of disagreeing nearest-neighbors, W(x, σ t ) = 1 1 1 2 2d σ t(x) σ t (y) (1) y N (x) N (x) denoting the set of all nearest-neighbor sites of x.

Definition of the model The transition rate vanishes when there are no nearest-neighbors sites with opposite opinion, hence the model is characterized by absence of bulk noise. The model has two trivial extremal stationary distributions, the point-masses δ 0 and δ 1 on σ 0 and σ 1 respectively, which represent full consensus. But are there steady states in which there is the coexistence of an infinite number of both opinion values? On the other hand, in which conditions does clustering lead to full consensus? More precisely, if for all x,y Z d we will say that coarsening occurs. lim Prob{σ t(x) σ t (y)} = 0 (2) t + Alessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 6 / 34

Alessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 7 / 34 Master equation The probability distribution P(σ) of the process satisfies the following master equation d dt P(σ t) = ( ) W x (σt x )P(σt x ) W x (σ t )P(σ t ) x Z d (3) with σ x { 1, 1} Zd denoting that configuration of the system which differs from σ only in the site x. The linear structure of the spin-flip rate has the nice consequence that the equations for correlation functions of different orders decouple. In particular explicit analytical solutions for single-body and two-body correlation functions can be found for arbitrary dimension d.

Alessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 8 / 34 Correlation functions The single and two-body correlation functions satisfy d dt σ t(x) = = 1 2d x σ t (x) (4) d dt σ t(x)σ t (y) = 1 ) ( x + y σ t (x)σ t (y) (5) 2d with x denoting the discrete Laplace operator, x f (x) = 2d f (x) + d i=1 [f (x + e i) + f (x e i )] {e i } i=1,...,d denoting the set of unit vectors defining the lattice.

Alessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 9 / 34 Average site magnetization In the case of an initial configuration with a +1 voter sitting at the origin and surrounded by a sea of undecided voters, the asymptotic behaviour of the average site magnetization for t + is given by ( 2πt σ t (0) d ) d 2 (6) The opinion of a single voter relaxes to the average undecided opinion of the rest of the population.

Alessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 10 / 34 Two-body correlation function Regarding the two-body correlation function G(r, t) = σ t (r)σ t (0) by adopting a continuum limit and a quasi-static approximation one finds the following asymptotic behaviour G(r, t) 1 r t if d = 1 and 0 < r < t ( ) 2 ln (t/2) ln r if d = 2 and 1 < r < t/2 t/2 1 r d 2 if d > 2 and r > 1 in the case of totally uncorrelated initial condition. In the case d = 2, the magnitude of G(r, t) decays logarithmically and there is a typical growing length l(t) t 1/z, with z = 2 (7)

lessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 11 / 34 Fraction of active interfaces To decide wether coarsening occurs or not, we look at the fraction of active bonds, defined as ( ) ρ + (t) = 1 1 d G ei (t) 2d i=1 (8) Using the results for the two-body correlation function, one finds t 1 2 if d = 1 ρ + (t) ln 1 t if d = 2 as t + a b t d/2 if d > 2 (9)

Coarsening vs Coexistence Then two different asymptotic behaviours are expected depending on the dimensionality of the lattice : for d 2, the probability that two voters at a given separation had opposite opinion vanishes asymptotically, no matter how much distant they are, and coarsening eventually leads to a single-domain final state ; for d > 2, an infinite system reaches a dynamics frustrated state, where opposite-opinion voters coexist and continually evolve in such a way that the average concentration of each type of voters remains fixed. Dimension d = 2 is particular since it lies at the border between the two cases. There is a coarsening process which brings the system towards the single-domain state, but it is very slow, since the density of active interfaces vanishes only as 1/ ln t. Alessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 12 / 34

Alessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 13 / 34 Average magnetization per site Remark Define the average magnetization per site as m(t) = lim L L d x L/2 σ t (x). m(t) is a conserved quantitity (to see this, just sum both sides of the eq. for σ t (x) over all sites of the lattice) Assume that the initial configuration is totally uncorrelated, with a fraction ρ of +1 voters, and suppose that the system reaches consensus in which the state of magnetization m = 1 occurs with probability p +1 (ρ). Then by conservation of m(t), one has p +1 (ρ) = ρ.

Duality There is an equivalence between the voter model and a system of coalescing random walks on the integer lattice Z d. Reconsider the problems in terms of fermionic occupation number, i.e. σ(x) η(x) = 1 2 (1 + σ(x)). Consider a collection of n continous-time rate-1 symmetric random walkers starting from sites x 1,..., x n and such that when two of them meet they coalesce. Then η t (x 1 )... η t (x n ) = E [ρ N(t)] with ρ the initial fraction of occupied sites, N(t) the number of random walks surviving at time t and E [ ] the average over all possible realizations of the coalescing random walks system. Alessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 14 / 34

Duality For the case n = 2 and totally uncorrelated initial condition with equal probability of having σ(x) = ±1 for any x Z 2, one has Prob{η t (x) η t (y)} = 1 2 [1 p h (x y, 2t)] (10) where p h (x 0, t) is the probability for a random walk whose initial position is x 0 to first hit the origin at time s t. in dimension d 2 random walks are recurrent, thus lim t + Prob{η t (x) η t (y)} = 0, then the systems coarsens until reaching consensus; in dimension d > 2 random walks are transient, hence lim t + Prob{η t (x) η t (y)} = const. > 0, and the system is characterized by an infinite family of steady-states in which the two opinions coexist. Alessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 15 / 34

lessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 16 / 34 Simulation of 2d case We focused on the case d = 2 and investigated the coarsening dynamics of the model, in order to compare it with the analogous properties of the 2dIM quenched at zero temperature. (a) t = 4 (b) t = 64 (c) t = 512 (d) t = 4096 2dIM (e) t = 4 (f) t = 64 (g) t = 512 (h) t = 4096 2dVM Figure : Snapshots of the Ising and voter models on a 2d square lattice with linear size L = 640. The working temperature for the Ising model is T = 0.

Alessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 17 / 34 Simulation of 2d case (maybe show movie here)

Alessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 18 / 34 Domain area dynamics For curvature-driven coarsening processes, such as domain growth in the zero-temperature 2d Glauber-Ising dynamics, after a sufficiently large time, all interfaces move with a local velocity which is proportional to the local curvature and points in the direction of decreasing the curvature itself. Interfaces tend to disappear independently of one another. (maybe put image here to explain)

Alessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 19 / 34 Domain area dynamics By using a dynamical scaling hypothesis or relying only on Allen-Cahn equation and making suitable approximations, one finds that the number density of domain areas for the 2dIM, quenched from infinite temperature to T = 0, has the scaling form n d (A, t) 2c d (λ d t) τ 2 (A + λ d t) τ = 2c d A 2 (λ d t/a) τ 2 [1 + λ d t/a] τ (11) with τ a characteristic exponent related to the critical percolation state that the system reaches early in the dynamics, 2c d (τ 1)(τ 2) and λ d a phenomenological parameter with the dimension of a diffusion constant which sets the growth rate of the average domain area.

Alessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 20 / 34 Scaling hypothesis For the 2d voter model the dynamics is led solely by interfacial noise The structure of domains is very different from the one of the zero-temperature 2dIM: they in fact preserve their fractal geometry at all scales even at late stages of the dynamics. Nevertheless, we suppose that also in the case of the 2dVM the domain area distribution satisfies a similar scaling law, n d (A, t) 1 ( ) A A ν Φ (λt) α (12)

Scaling hypothesis We expect two different scales domains of areas A < (λt) α have already the statistics of the asymptotic regime of the dynamics, with exponent ν related to critical percolation. n d (A, t) c A ν (13) domains of areas A (λt) α are not yet equilibrated to the asymptotic percolating regime, but still the distribution n d is self-similar at different times if areas are rescaled by a typical growing area A(t) (λt) α. Alessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 21 / 34

Domain area distribution n d (A, t) 10 1 10 3 10 5 10 7 10 9 10 11 1 10 10 2 10 3 10 4 10 5 10 6 A t = 0 2 8 32 128 512 2048 8192 16384 Figure : Domain area number density for the 2dVM on a finite square lattice of linear size L = 640, evolving in time from a random initial condition. At t = 0 the distribution decays very quicly for large values of A; as time elapses, a power law extending over several decades develops; for large times, a bump with support over areas that are of the order of magnitude of the size of the system also appears. Alessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 22 / 34

Determination of exponent ν We tried to determine the exponent ν characterizing the asymptotic behaviour of the distribution n d by fitting the data corresponding to a finite lattice of linear size L = 640 at a time t 16 10 3. A ν n d (A, t) 1 10 1 10 2 1 10 10 2 10 3 10 4 10 5 10 6 10 7 A t = 0 2 8 32 128 512 2048 4096 16384 c Figure : n d (A, t) A ν plotted against A for a lattice of size L = 640. Alessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 23 / 34 ν 1.98 c 0.022 The plateau emerging at the latest time shown, corresponds to that range of areas whose statistics has already reached the asymptotic form.

Alessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 24 / 34 Testing the dynamical scaling hypothesis A τ nd(a, t) 1 10 1 10 2 t = 64 128 256 512 1024 2048 4096 8192 16384 10 3 10 2 10 1 1 10 10 2 10 3 A/t α Figure : n d (A, t) A ν plotted against the rescaled area A/t α for a lattice of size L = 640. Best collapse was obtained with α 1.19 Two different behaviours: for small values of t, n d (A, t) c A ν (A/t α ) a with 0 < a < ν (see the red dashed line); for large values of t, n d (A, t) c A ν, save deviations due to the bump.

Alessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 25 / 34 Testing the dynamical scaling hypothesis For intermediate times, the distribution of domain areas displays both behaviours depending on the scale. (insert plot of curve corresponding to intermediate time)

Alessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 26 / 34 Percolating regime on finite lattices In the case of the zero-temperature 2dIM the structure of the asymptotic percolating regime is decided early in the dynamics: about 8 Monte Carlo simulation steps are enough for a stable pattern of more than one percolating domains to establish. This pattern preserves its topology throughout the dynamics, eventually leading the system to freeze in a stripes final state. For the 2dVM coarsening leads to a full consensus state in any case, even if the process is very slow (the fraction of active bonds decays only logarithmically). So, is it possible to have a relatively long-lived metastable state in which the system shows a pattern of stable percolating domains?

Alessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 27 / 34 Percolation time t P (L) For the 2dIM the blocked percolating state is reached in a typical time t P which depends on the lattice size L as t P (L) L α P (14) with α P 0.5, as found by Blanchard et al. [?]. Blanchard-et-al To compute t P we looked at the time evolution of the number of percolating domains N P (t; L), assumming the following scaling law ( ) t N P (t; L) N t P (L) (15)

Alessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 28 / 34 Percolation time t P (L) 1.2 1 0.8 N P (t) 0.6 0.4 0.2 0 10 3 10 2 10 1 1 t/l αp L = 20 L = 40 L = 80 L = 160 Figure : Average number of wrapping domains N P (t) plotted against rescaled time t/l α P, for systems of sizes L = 20, 40, 80, 160. Best collapse was obtained with α P 1.67 Better estimates of α P by using larger systems were more difficult to obtain due to the huge amount of time required for the simulations.

Correction to scaling The introduction of an additional length scale L(L) = l(t P (L)) L α P /z yields a better rescaling of the correlation function, ( ) G(x, t; L) = 1 ln t g x l(t), L(L) l(t) 10 L = 80, t = 256.00 812.75 2580.32 8192.01 L = 80, t = 256.00 L = 160, t = 812.75 L = 320,t = 2580.32 L = 640, t = 8192.01 G(x, t) ln (t) 1 0.1 0 0.4 0.8 1.2 1.6 x/ t 0 0.4 0.8 1.2 1.6 2 x/ t Figure : In (a) G(x, t) ln t is plotted against scaled distance x/ t for L = 80. In (b) the lattice sizes L i and corresponding values of time t i were chosen such that the quantity L(L i )/l(t i ) ( L α P i /t i ) 1/z were a constant. Alessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 29 / 34

Alessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 30 / 34 Summary We investigated the general properties of the voter model dynamics and, in particular, tried to characterize the coarsening process in the two-dimensional case: the fraction of active bonds ρ + decays logarithmically possibility of a metastable state there is a characteristic growing length l(t) that scales as t 1/2 the domain area distribution ( seems to follow well the scaling law n d (A, t) 1 A A Φ ν (λt) ), with ν 2 and α 1.2 α the typical time t P required for the system to reach a stable percolating regime scales with the system size as t P L α P, with α P 1.67

Alessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 31 / 34 Summary Future perspectives: a full characterization of the geometrical properties of the spin clusters, such as the time-evolution of their fractal dimension and the relation between domain area and interface length, would give more insight into the coarsening process. theoretical predictions on the domain area dynamics are missing as far as we know: an attempt could be made by recasting the problem in terms of a continous reaction-diffusion model it would be interesting to consider variants of the update rule, to see in which circumstances phase-ordering occurs (e.g. majority rule voter model, social impact theory etc. )

Alessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 32 / 34 Aknowledgements Aknowledgements

Alessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 33 / 34 References References

The End Alessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 34 / 34