Voter Model with Nearest-Neighbour Dynamics

• Voter model with nearest-neighbour interactions is a stochastic system characterized by local imitation dynamics and non-equilibrium scaling behaviors.
• It exhibits exact scaling forms for correlation and response functions, with dynamics governed by a diffusion equation and Schrödinger invariance.
• The framework benchmarks ageing and universality in nonequilibrium statistical mechanics, highlighting the impact of dimensionality and thermal noise.

The voter model with nearest-neighbour interactions is a stochastic interacting particle system that serves as a central paradigm for modelling consensus formation and non-equilibrium critical dynamics in spatially extended systems. Each site of a $d$-dimensional lattice hosts a spin variable $\sigma_n = \pm 1$, and the underlying stochastic evolution is specified by local update events—nearest-neighbour imitation—without energy minimization or detailed balance. The scaling properties, correlation functions, and dynamical symmetries of the model are by now fully understood across arbitrary spatial dimension; recent exact results have established that all dynamical two-point functions and response functions exhibit scaling forms dictated by the Schrödinger group, with dynamical exponent $\mathpzc{z} = 2$ and dominant fluctuations sourced from the heat bath. This framework provides a rigorous non-equilibrium analog of the classic critical Ising model and identifies the upper critical dimension as $d^* = 2$.

1. Model Definition and Microscopic Dynamics

The voter model is defined by a field of Ising spins $\sigma_n(t) = \pm 1$ on each vertex $n$ of a $d$-dimensional regular lattice, with transitions governed by the following stochastic rule:

At each time step, a site $n$ is selected at random and adopts the value of a randomly chosen nearest neighbour $m$,

$\sigma_n \to \sigma_m$

with transition rate (per unit time)

$$w_n(\{\sigma\}) = \frac{1}{2d} \sum_{m \sim n} [1 - \sigma_n \sigma_m]$$

where the sum is over the $2d$ lattice nearest neighbours of $n$. This dynamics conserves the number of each opinion only globally in expectation, and does not satisfy detailed balance except in trivial (fully ordered) limits. The full master equation corresponds, in the continuum and large-scale limit, to a linear diffusion equation for local averages and correlation functions.

2. Exact Multi-time Correlators and Response Functions

The principal observables are the space-time two-point correlation function

$C(t; r) = \langle \phi(t, r) \phi(t, 0) \rangle$

and the two-time auto-correlation and linear response functions: $$C(t, s) = \langle \phi(t, 0) \phi(s, 0) \rangle, \qquad R(t, s; r) = \left. \frac{\delta \langle \phi(t, r)\rangle}{\delta h(s, 0)}\right|_{h=0} = \langle \phi(t, r) \widetilde{\phi}(s, 0) \rangle$$ where $h$ is a local, time-dependent external field and $\widetilde{\phi}$ is the standard response operator in the Janssen–de Dominicis formalism.

All correlators solve the $d$-dimensional diffusion equation: $\partial_t C(t; r) = \Delta_{r} C(t; r)$ with $\Delta_r = \partial_r^2 + (d-1) r^{-1} \partial_r$ for radial coordinates. The scaling solution, employing the ansatz $C(t; r) = t^{-b} F_C(u)$ with $u = r/\sqrt{t}$, yields for $d < 2$

$$C(t; r) = \frac{1}{\Gamma(1 - d/2)} \exp\left(- \frac{r^2}{4t} \right) U\left(\frac{d}{2}, \frac{d}{2}; \frac{r^2}{4t}\right)$$

where $U(a, b; z)$ is the Tricomi confluent hypergeometric function and $b = 0$. For $d > 2$, the stationary correlations require scaling with $b = (d - 2)/2$ and take the form

$$C(t; r) = \mathfrak{C}_0\, r^{2-d}\, \frac{\Gamma(\frac{d}{2} - 1, \frac{r^2}{4t})}{\Gamma(\frac{d}{2} - 1)}$$

where $\Gamma(a, z)$ is the incomplete gamma function.

The two-time auto-correlator displays exact aging scaling

$C(t, s) = s^{-b} f_C\left(\frac{t}{s}\right)$

with scaling function

$$f_C(y) = \frac{1}{\Gamma(1-\frac{d}{2}) \Gamma(\frac{d}{2} + 1)} \left(\frac{2}{y+1}\right)^{d/2} {}_2F_1\left(\frac{d}{2}, 1; \frac{d}{2}+1; \frac{2}{y+1}\right)$$

for $d < 2$, where ${}_2F_1$ is the Gaussian hypergeometric function; for $d > 2$ a multiplicative factor $s^{d/2-1}$ applies. The large-$y$ asymptotics yield

$C(t, s) \sim (t/s)^{-d}$

with autocorrelation decay exponent $\lambda = d$.

The two-time linear response function is

$$R(t, s; r) = \mathfrak{R}_0 s^{-1-a} \left(\frac{t}{s}-1\right)^{-d/2} \exp\left[-\frac{r^2}{t-s}\right]$$

with aging exponent $a = 0$ for $d < 2$ and $a = d/2 - 1$ for $d > 2$, and dynamical exponent $\mathpzc{z} = 2$.

3. Schrödinger Invariance and Dynamical Scaling

At long times and large scales, the voter model's evolution equations for correlation and response are invariant under the non-relativistic Schrödinger group. The defining equation,

$(2\mathcal{M}\partial_t - \Delta_r) \phi(t, r) = 0$

with effective “mass” $\mathcal{M} = 1$, admits the Schrödinger algebra as a local Lie symmetry: $$X_{-1} = -\partial_t, \quad X_{0} = -t\partial_t - \frac{1}{2} r\partial_r - \delta, \quad X_{1} = -t^2\partial_t - t r\partial_r - \frac{\mathcal{M}}{2} r^2 - 2\delta t$$ plus space translations and Galilei boosts. Consequences include:

• The emergence of dynamical scaling, $r \sim t^{1/\mathpzc{z}}$ with $\mathpzc{z} = 2$.
• Universal forms for correlation and response scaling functions determined (up to non-universal prefactors and scaling dimensions $\delta$) by the representation theory of the Schrödinger algebra.
• The explicit forms above for $C(t; r)$, $C(t, s)$, and $R(t, s; r)$ all satisfy Schrödinger invariance, matching the generic prediction

$$R(t, s; r) = \Theta(t-s)\, (t-s)^{-2\delta}\, \exp\left[-\frac{\mathcal{M} r^2}{2(t-s)}\right]$$

and for ageing, scaling forms

$C(t, s; r) = s^{-b} F_C\left(\frac{t}{s}; \frac{r}{s^{1/\mathpzc{z}}}\right), \qquad R(t, s; r) = s^{-1-a} F_R\left(\frac{t}{s}; \frac{r}{s^{1/\mathpzc{z}}}\right)$

with exponents $a$ and $b$ set by $d$.

4. Ageing, Criticality, and Universality Class

Ageing in the voter model refers to the explicit dependence of two-time observables on the ratio $t/s$, with $t > s$, and is a hallmark of non-equilibrium critical dynamics. The model sits at a non-equilibrium critical point (with two absorbing states), but exhibits universal scaling determined entirely by the bath (external) noise, in contrast to models such as the Ising or Potts models below criticality where domain coarsening is curvature-driven. In the Janssen–de Dominicis field theory, the quadratic Gaussian action (diffusion kernel) and thermal noise structure are sufficient to fix the scaling exponents and forms. The upper critical dimension is $d^* = 2$; for $d < 2$ coarsening (domain growth) dominates, while for $d > 2$ stationary critical correlations prevail.

5. Diagrams and Analytical Forms

Key analytical expressions include:

• Single-time correlator, $d < 2$:

$$C(t; r) = \frac{1}{\Gamma(1-\frac{d}{2})} \exp\left(-\frac{r^2}{4t}\right) U\left(\frac{d}{2}, \frac{d}{2}; \frac{r^2}{4t}\right)$$

• Two-time correlator, $d < 2$:

$$C(t,s) = s^{-b} \frac{1}{\Gamma(1-\frac{d}{2})\Gamma(\frac{d}{2}+1)} \left(\frac{2}{y+1}\right)^{d/2} {}_2F_1\left(\frac{d}{2}, 1; \frac{d}{2}+1; \frac{2}{y+1}\right)$$

• Two-time response, $d < 2$:

$$R(t, s; r) = \mathfrak{R}_0 s^{-1} \left(\frac{t}{s}-1\right)^{-d/2} \exp\left(-\frac{r^2}{t-s}\right)$$

Representative plots (see Figs. 1–4 in (Henkel et al., 15 Sep 2025)) show data collapse for $C(t; r)$ and $C(t,s)$ versus $(r/\sqrt{t})$ and $(t/s)$ across different $d$, confirming the theoretical scaling forms and the change in scaling regime at $d=2$.

6. Significance in Nonequilibrium Statistical Mechanics

The voter model, by virtue of its exact scaling functions and symmetry-determined forms, sets a benchmark for universality in non-equilibrium critical dynamics. The dominance of heat bath noise in determining fluctuations, the emergence of Schrödinger invariance, and the sharp distinction between $d < 2$ and $d > 2$ regimes stand in contrast to equilibrium models with surface-tension-driven ordering. This analysis provides a quantitative underpinning for the ageing classification and for the universality of two-time functions in a broad family of stochastic models, including but not restricted to the voter model.

In conclusion, Schrödinger invariance controls the long-time, large-scale dynamics of the voter model with nearest-neighbour interactions in all dimensions $d > 0$, yielding universal, symmetry-fixed correlation and response scaling functions with critical exponents dependent only on $d$ and the noise source, and marking the model as a canonical example of non-equilibrium criticality without detailed balance (Henkel et al., 15 Sep 2025).