Symmetric Noisy Voter Model

• The symmetric noisy voter model is a continuous-time stochastic process with unbiased state transitions driven by both neighbor imitation and random noise.
• It employs probabilistic rules and analytical tools like the master equation and Fokker–Planck approximation to quantify mixing times, finite-size effects, and dynamical transitions.
• The model reveals critical scaling laws and phase transitions, offering insights into consensus dynamics and ergodicity across applications in genetics, opinion dynamics, and statistical physics.

The symmetric noisy voter model is a paradigmatic stochastic process for collective dynamics incorporating both imitation (copying) and individualistic spontaneous change (noise), with complete symmetry among all admissible states. Originally developed for applications in population genetics, opinion dynamics, and statistical physics, the symmetric noisy voter model generalizes the classical voter model by replacing strictly deterministic copying with probabilistic, noise-induced imperfect copying or random updates. Its rigorous formulation, stationary properties, dynamical transitions, and finite-size effects reveal a rich phenomenology, including characteristic scaling behaviors, mixing cutoffs, and universality across related models.

1. Formal Definition and Dynamics

The symmetric noisy voter model is formulated as a continuous-time Markov process on $N$ agents (sites), each in one of $S$ (possibly two) equivalent states, denoted $\theta_k$ (e.g., spin states: $k=0,1,\ldots,S-1$ with $\theta_k=2\pi k/S$). The updating dynamics are defined on either complete graphs (mean-field) or general graphs with adjacency relations.

Update Mechanisms:

• Imitation: With probability $1-p$, a focal agent $i$ selects one or multiple neighbors (e.g., $q=1$ for voter, $q>1$ for $q$-voter) and, if a specified unanimity or majority rule is satisfied, adopts their state. In the basic voter model $(q=1)$, $i$ copies the state of a randomly chosen neighbor.
• Noise: With probability $p$ (or rate $a$), an agent replaces its state by a choice independently drawn from the uniform distribution over all $S$ states, or perturbs the copied value by a random error drawn from a symmetric distribution (e.g., uniform in $[-\pi\eta,\pi\eta]$ for continuous states).

Symmetry:

All states are a priori equivalent (e.g., $Z_S$ symmetry), and the noise is unbiased (no preference for any state).

In the binary case $(S=2)$ and on a finite undirected graph $G$, the generator for the dynamics is

$$\mathcal{L}f(\eta) = \sum_{x\in V} \left\{ \theta \cdot \frac{1}{2} \left[ f(\eta^{x \to 0}) + f(\eta^{x \to 1}) - 2f(\eta) \right] + (1-\theta) \frac{1}{\deg(x)} \sum_{y\sim x} [f(\eta^{x \to \eta(y)}) - f(\eta)] \right\}$$

where $\eta$ is the configuration, $\theta$ the noise parameter, and $\eta^{x \to y}$ the configuration obtained by setting $\eta(x)=y$ (Caddeo et al., 22 Jul 2025).

The multi-state angular version with imperfect copying and noise amplitude $\eta$ is defined by, at each time step, selecting a random agent $i$, a target $j$, and setting

$$\Theta_i(t+\delta t) = \Theta_j(t) + \xi_i, \qquad \xi_i \sim \mathrm{Unif}(-\pi\eta, \pi\eta)$$

for angular states $\theta_k$, yielding imperfect copying (Vazquez et al., 2019).

2. Stationary Distribution and Ergodicity

For any symmetric noise amplitude $\theta=a=b>0$ in the binary case, the stationary distribution is the uniform product measure over all $2^N$ configurations on the complete graph, or the uniform measure within each equivalence class for the $S$-state model (Cox et al., 2014, Caddeo et al., 22 Jul 2025, Cox et al., 2012). On the complete graph, the empirical distribution of states converges to the multinomial (or in the large-$N$ limit, the Dirichlet or Beta in the binary case):

• The stationary law of the empirical state density $x\in[0,1]$ under symmetric noise is Beta$(a, b)$ in the $N\to\infty$ limit (Aljovin et al., 9 Sep 2024).
• For the discrete multi-angle model, the uniform (disordered) configuration is unique in the presence of any noise, with $\langle x_k\rangle = 1/S$ for all $k$ (Vazquez et al., 2019).

In infinite (transitive) lattices, ergodicity holds universally for $\theta>0$ (or $\epsilon>0$ in alternative notation), ensuring convergence to a spatial product measure with each site marginally $1/S$ (Cox et al., 2012).

3. Order Parameter and Scaling of Disorder

The level of global order, or consensus, is quantified via suitable order parameters:

• Angular Model:

$$\psi(t) = \left| \frac{1}{N} \sum_{m=1}^N e^{i\Theta_m(t)} \right|^2$$

so that $\psi=1$ for consensus and $\psi=0$ for complete disorder (Vazquez et al., 2019).

• Multi-state Model:

$$C(t) = \max_{i} x_i(t), \quad m(t) = \frac{S C(t)-1}{S-1}$$

$m=1$ for consensus, $m=0$ for uniformity (Nowak et al., 2020).

Thermodynamic Limit and Noise Scaling:

• In the symmetric multi-angle model (Vazquez et al., 2019):
  • For perfect copying $(\eta=0)$, $\langle \psi \rangle = 1$ (perfect order) for all $N$.
  • For any $\eta>0$,

  $$\langle \psi \rangle \simeq \frac{6}{\pi^2 \eta^2 N} \quad (\eta \ll 1, \, \eta^2 N \gtrsim 1)$$

  and the system is totally disordered as $N \to \infty$. - For vanishing $\eta$ but fixed $N$,

  $$\langle \psi \rangle \simeq 1 - 1.64 \eta^2 N \quad (\eta \to 0,\, \eta^2 N \ll 1)$$

Thus, global order is destroyed for any finite noise as $N\to\infty$.

• In non-linear extensions, true order–disorder phase transitions (continuous or discontinuous) can arise for non-linear herding exponents $(\alpha>1)$ (Peralta et al., 2018, Llabrés et al., 16 May 2025).

4. Relaxation, Mixing, and Cutoff Phenomena

The relaxation dynamics exhibit mixing times and cutoff phenomena that depend on the graph structure and noise magnitude:

• Complete Graph: Under symmetric noise, convergence of the empirical density to its stationary law (Kantorovich/Wasserstein-1 distance) is smooth with a system-size dependent mixing time $O(N)$—i.e., there is no cutoff (Aljovin et al., 9 Sep 2024).

• Site Thermalization: The process by which the system forgets the initial positions of states (but not their total number) displays a cutoff at time $(1/2)\log n$ with $O(1)$ window, corresponding to the thermalization of particle labels (Aljovin et al., 9 Sep 2024, Cox et al., 2014, Ramadas, 2014).

• Finite Graphs: For $n$ sites with noise $\theta$, the cutoff for mixing in total variation distance is at $t_n = (1/(2\theta)) \log n$, with the window of order $O(1)$. This is tight both for worst-case and typical initial conditions (Cox et al., 2014, Caddeo et al., 22 Jul 2025). Fastest-mixing initial states (alternating or rainbow colorings) achieve strictly faster relaxation in certain graph classes (Caddeo et al., 22 Jul 2025).

• General Graphs: On bounded-degree graphs of subexponential growth, cutoff at $(1/(2\theta))\log n$ holds, with initial-condition dependence for more complex state spaces (Caddeo et al., 22 Jul 2025).

The model always remains in the high-temperature, strong-mixing regime for any nonzero symmetric noise, characterized by exponential temporal and spatial mixing and absence of absorbing states (Ramadas, 2014).

5. Phase Transitions and Universality

For the linear binary symmetric noisy voter, the only phase transition is trivial: for $\theta=0$, the system has absorbing consensus states; for any $\theta>0$, ergodicity is restored, and the stationary measure is fully disordered. However, when generalizing to non-linear interaction rules or multi-state models, true phase transitions emerge:

• Nonlinear Noisy Voter Model:

  • For $\alpha > 1$ (herding non-linearity), the model exhibits a second-order (Ising-like) continuous phase transition at a critical noise level $\epsilon_c = 2^{-\alpha} (\alpha-1)$ in the $N\to\infty$ limit; for $\alpha=1$, only a finite-size-induced transition occurs (Peralta et al., 2018, Llabrés et al., 16 May 2025).
  • The continuous transition line (Ising) and the discontinuous Modified Generalized Voter (MGV) transition line meet at a tricritical point. The critical exponents coincide with Landau mean-field or 2D Ising universality classes depending on dimensionality (Llabrés et al., 16 May 2025).
• Noisy $q$-Voter Models:
  • For $s>2$ states, discontinuous (first-order) order–disorder transitions are present for all $q>1$ (first-order even for $q=2$), both under annealed and quenched disorder, in contrast to binary cases where the transition is continuous for small $q$ (Nowak et al., 2020).
  • Critical points are given by

  $$p_c^{\rm ann}(q,s) = \frac{q-1}{q-1 + s^{q-1}}, \quad p_c^{\rm quen}(q,s) = \frac{q-1}{q}$$ - The phase diagrams exhibit hysteresis and coexistence depending on disorder type.

• Finite-size Scaling and Network Effects:

  • Nonlinear noisy voter models display universal finite-size scaling laws for moments of the order parameter, with scaling functions depending on effective system size and network heterogeneity (e.g., $$N_{\rm eff} = N \langle k \rangle^2/\langle k^2 \rangle$$ on complex networks) (Carro et al., 2016, Peralta et al., 2018).
  • Degree heterogeneity in the underlying network shifts critical points and modifies the autocorrelation structure, enabling network inference from macroscopic time series.

6. Analytical Solvability and Mean First Passage Times

Under mean-field (complete-graph) and diffusion scaling, the stationary and transient properties of the symmetric noisy voter model are analytically tractable:

• Master Equation and Fokker–Planck Approximation:

The exact master equation on configuration space can be approximated by a Fokker–Planck or Langevin equation for the macroscopic order parameter (magnetization $m$ or angle density $x$), with explicit drift and diffusion coefficients capturing both drift toward disorder and finite-size effects (Vazquez et al., 2019, Peralta et al., 2018, Llabrés et al., 16 May 2025).

• Mean First Passage Times (MFPT):
  • For the process $dx = \varepsilon (1-2x) dt + \sqrt{2x(1-x)} dW_t$, the MFPT from any $x_0$ to boundaries $L,H$ satisfies a second-order ODE with closed-form solutions in terms of incomplete beta and Meijer $G$-functions (Kazakevičius et al., 2 Dec 2025).
  • MFPT exhibits boundary-induced asymmetry unless boundaries are symmetrically placed about $x=1/2$; symmetry is otherwise restored only in that special case.
  • In the large-noise (Kramers) regime, MFPT grows as $$\sim \varepsilon^{-3/2} \exp (\varepsilon \Delta U)$$, consistent with Arrhenius-type escape scaling.

7. Duality, Complete Convergence, and Ergodicity

The symmetric noisy voter model can be analyzed using duality to particle systems with annihilation and branching (Cox et al., 2012). For any symmetric noise amplitude, all initial conditions converge to a unique ergodic stationary law, and the system cannot be trapped in absorbing states. Dual processes allow for mathematical proofs of convergence via block-construction and percolation arguments, applicable on $\mathbb{Z}^d$ and more general graphs. The only critical point is $\theta=0$, at which the model reduces to the classical voter model with its well-understood clustering properties.

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Key references:

• Multi-angle noisy voter, order parameter scaling, and Monte Carlo: (Vazquez et al., 2019)
• Nonlinear and $q$-voter transitions, universality, and phase diagrams: (Nowak et al., 2020, Llabrés et al., 16 May 2025, Peralta et al., 2018)
• Discrete-time cutoff, mixing, and thermalization: (Cox et al., 2014, Aljovin et al., 9 Sep 2024, Caddeo et al., 22 Jul 2025)
• Mean first passage time, boundary asymmetry, and Kramers' law: (Kazakevičius et al., 2 Dec 2025)
• Infinite graphs, ergodicity, and duality: (Cox et al., 2012)
• Network effects and macroscopic inference: (Carro et al., 2016)

These results collectively provide a comprehensive macroscopic and microscopic characterization of the symmetric noisy voter model and its extended variants.