Persistence in a Random Bond Ising Model of Socio-Econo Dynamics
📝 Abstract
We study the persistence phenomenon in a socio-econo dynamics model using computer simulations at a finite temperature on hypercubic lattices in dimensions up to 5. The model includes a social\rq local field which contains the magnetization at time $t $. The nearest neighbour quenched interactions are drawn from a binary distribution which is a function of the bond concentration, $p $. The decay of the persistence probability in the model depends on both the spatial dimension and $p $. We find no evidence of blocking\rq in this model. We also discuss the implications of our results for possible applications in the social and economic fields. It is suggested that the absence, or otherwise, of blocking could be used as a criterion to decide on the validity of a given model in different scenarios.
💡 Analysis
We study the persistence phenomenon in a socio-econo dynamics model using computer simulations at a finite temperature on hypercubic lattices in dimensions up to 5. The model includes a social\rq local field which contains the magnetization at time $t $. The nearest neighbour quenched interactions are drawn from a binary distribution which is a function of the bond concentration, $p $. The decay of the persistence probability in the model depends on both the spatial dimension and $p $. We find no evidence of blocking\rq in this model. We also discuss the implications of our results for possible applications in the social and economic fields. It is suggested that the absence, or otherwise, of blocking could be used as a criterion to decide on the validity of a given model in different scenarios.
📄 Content
The persistence problem is concerned with the fraction of space which persists in its initial (t = 0) state up to some later time t. The problem has been extensively studied over the past decade for pure spin systems at both zero [1][2][3][4] and non-zero [5] temperatures.
Typically, in the non-equilibrium dynamics of spin systems at zero-temperature, the system is prepared initially in a random state and the fraction of spins, P (t), that persists in the same state as at t = 0 up to some later time t is studied. For the pure ferromagnetic Ising model on a square lattice the persistence probability has been found to decay algebraically [1][2][3][4]
where θ ∼ 0.22 is the non-trivial persistence exponent [1][2][3]. Derrida et al [4] have shown analytically that for the pure 1d Ising model θ = 3/8. The actual value of θ depends on both the spin [6] and spatial [3] dimensionalities; see Ray [7] for a recent review.
At criticality [5], consideration of the global order parameter leads to a value of θ global ∼ 0.5
for the pure two-dimensional Ising model.
It has been only fairly recently established that systems containing disorder [8][9][10] exhibit different persistence behaviour to that of pure systems. A key finding [8][9]11] is the appearance of ‘blocking’ regardless of the amount of disorder present in the system. ‘Blocked’ spins are effectively isolated from the behaviour of the rest of the system in the sense that they never flip. As a result, P (∞) > 0 and the key quantity of interest is the residual persistence given by
Note that for the five dimensional pure Ising model without any disorder blocking has also been observed at T = 0 [3]. At finite temperature there is no evidence of blocking [12].
As well as theoretical models, the persistence phenomenon has also been studied in a wide range of experimental systems and the value of θ ranges from 0.19 to 1.02 [13][14][15], depending on the system. A considerable amount of the recent theoretical effort has gone into obtaining numerical estimates of θ for different models [1][2][3][4][5][6][7][8][9][10][11]. Recently, it has been found that the behaviour of the random Ising ferromagnet at zero temperature on a Voronoi-Delaunay lattice [16] is very similar to the behaviour on the diluted ferromagnetic square lattice [8,9].
In this work we add to the knowledge and understanding regarding persistence by presenting the simulation results for the behaviour of a recently proposed spin model which appears to reproduce the intermittency observed in real financial markets [17]. In the next section we discuss the model in detail. In Section III we give an outline of the method used and the values of the various parameters employed. Section IV describes the results and the consequent implications for using the model in a financial or social context. Finally, in Section V there is a brief conclusion.
Yamano [16] has proposed a ‘minimalist’ version of the Bornholdt model [18]. We study the persistence phenomenon in the former. In this model one has N market traders, denoted by Ising spins S i (t), i = 1 . . . N, located on the sites of a hypercubic lattice, L d = N. The action of the i th trader of buying or selling a share of a traded stock or commodity at time step t corresponds to the spin variable S i (t) assuming the value +1 or -1, respectively. Hence, at each time step, a given trader will be either buying or selling. A local field, h i (t), determines the dynamics of the spins and, hence, the action of the trader. We follow [16] and assume that
where the first summation runs over the nearest neighbours of i only, α > 0 is a parameter coupling to the absolute magnetization. The nearest neighbour interactions are selected randomly from
where p is the concentration of ferromagnetic bonds. The case p = 1/2 corresponds to the ±J Edwards-Anderson spin-glass [11]. Each agent is updated according to the following heat bath dynamics:
where q is the probability of updating and T is temperature. The first term on the right hand side in equation ( 3) contains the influence of the neighbours and the second term reflects the external environment. The balance between the two terms determines whether an agent buys or sells. If h i (t) = 0, the agent is equally likely to buy or sell as q = 1/2.
If, however, h i (t) > 0, then q > 1/2 and agent i is more likely to buy than sell. Similarly, if h i (t) < 0, then we have q < 1/2 and the agent is more likely to sell than buy. α and T are tunable parameters in our model. The values we select for these are determined by the requirement that the model should be able to reproduce, at least qualitatively, some aspect of actual behaviour observed in a real market. In this model the return is defined in terms of the logarithm of the absolute value of the magnetization,
A key stylised fact observed in real financial markets is the intermittent or ‘bursty’ behaviour in the returns [19]. Simulations [16] in spatial dimensions rang
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