Schrödinger-invariance in the voter model (2509.11654v1)

Abstract: Exact single-time and two-time correlations and the two-time response function are found for the order-parameter in the voter model with nearest-neighbour interactions. Their explicit dynamical scaling functions are shown to be continuous functions of the space dimension $d>0$. Their form reproduces the predictions of non-equilibrium representations of the Schr\"odinger algebra for models with dynamical exponent $\mathpzc{z}=2$ and with the dominant noise-source coming from the heat bath. Hence the ageing in the voter model is a paradigm for relaxations in non-equilibrium critical dynamics, without detailed balance, and with the upper critical dimension $d^*=2$.

• The paper establishes that the voter model's scaling functions for correlation and response are exactly fixed by Schrödinger-invariance, linking non-equilibrium dynamics to representations of the Schrödinger algebra.
• It reveals distinct scaling regimes for d<2 and d>2, with logarithmic corrections at d=2, thereby highlighting the critical role of spatial dimensionality in ageing dynamics.
• A field-theoretic approach employing the Janssen-de Dominicis formalism connects composite operators to the model's universal ageing behavior, offering insights applicable to complex systems and AI.

Schrödinger-Invariance and Ageing in the Voter Model

Introduction and Motivation

The paper "Schrödinger-invariance in the voter model" (2509.11654) presents a comprehensive analytical paper of the voter model, focusing on its non-equilibrium ageing dynamics and the emergence of Schrödinger-invariance in its scaling functions. The voter model, a paradigmatic example of a non-equilibrium spin system without detailed balance, is analyzed in arbitrary spatial dimension $d>0$, including non-integer values. The work provides exact results for single-time and two-time correlation and response functions, elucidating their scaling forms and connecting them to representations of the Schrödinger algebra with dynamical exponent $\mathpzc{z}=2$.

The Voter Model and Ageing Dynamics

The voter model is defined on a $d$-dimensional hypercubic lattice with spin variables $\sigma_{\mathbf{n}} = \pm 1$ and evolves via single-spin flip dynamics with transition rates determined by the local neighborhood. Unlike the Glauber-Ising model, the voter model lacks detailed balance (except in $d=1$), leading to two absorbing states and a critical line separating non-critical phases. The model is analytically tractable in arbitrary dimension and serves as a testbed for non-equilibrium statistical mechanics, including applications in opinion dynamics and network theory.

Physical ageing in the voter model is characterized by three properties: (I) slow, non-exponential relaxation, (II) breaking of time-translation invariance, and (III) dynamical scaling. The system is typically initialized in a fully disordered state, and observables of interest include the single-time correlator $C(s; r)$, the two-time correlator $C(t, s; r)$, and the two-time response function $R(t, s; r)$. The analysis focuses on the scaling forms of these quantities in the long-time, large-distance regime.

Exact Results for Correlation and Response Functions

Single-Time Correlator

The single-time correlator $C(t; r)$ satisfies a diffusion equation in the continuum limit. The scaling ansatz $C(t; r) = t^{-b} F_C(r t^{-1/2})$ leads to a differential equation for $F_C$, whose solution involves confluent hypergeometric functions. The exponent $b$ and the scaling function depend continuously on $d$:

• For $d<2$, $b=0$ and $C(t; r)$ saturates at $r=0$ with a $d$-dependent approach, violating Porod's law except for $d=1$.
• For $d>2$, $b=(d-2)/2$ and $C(t; r)$ exhibits a power-law decay at small $r$, reminiscent of mean-field critical behavior.
• At $d=2$, logarithmic corrections appear, and the scaling form is modified by a $1/\ln t$ prefactor.

Two-Time Correlator

The two-time correlator $C(t, s; r)$ is obtained by solving a diffusion equation with initial condition set by the single-time correlator. The scaling form $C(t, s; r) = s^{-b} F_C(t/s, r/s^{1/2})$ is confirmed, with explicit expressions in terms of hypergeometric functions. The auto-correlation exponent $\lambda_C = d$ for all $d \neq 2$, and the scaling function exhibits:

• Saturation for $y = t/s \to 1$ when $d<2$.
• Power-law behavior in $y-1$ for $d>2$.
• Logarithmic corrections at $d=2$.

Two-Time Response Function

The response function $R(t, s; r)$ is derived via a perturbation in the master equation, leading to a Gaussian form in the scaling variable. The scaling exponents $a$ and $a'$ are determined, and the response function matches the predictions of Schrödinger-invariance. The equality $\lambda_R = \lambda_C = d$ holds, and the scaling forms are consistent with mean-field theory for $d>2$.

Schrödinger-Invariance and Dynamical Symmetries

A central result of the paper is the demonstration that the scaling functions of the voter model are fully determined by non-equilibrium representations of the Schrödinger algebra. The analysis employs a field-theoretic approach based on the Janssen-de Dominicis formalism, with the deterministic part of the action exhibiting Schrödinger symmetry. The mapping to non-equilibrium observables is achieved by a change of representation in the Lie algebra generators, introducing additional scaling dimensions $(\delta, \xi)$ for the scaling operators.

The explicit forms of the two-time response and correlation functions are shown to coincide with those predicted by Schrödinger-invariance, up to a finite set of scaling parameters. For $d=2$, a non-standard representation involving logarithmic scaling is required, which is constructed and shown to reproduce the exact results. The analysis confirms that the universal scaling functions are fixed by the properties of the composite response operator $\tilde{\phi}_2 = \tilde{\phi}^2$, and that the voter model falls into the universality class of non-equilibrium critical dynamics with $\mathpzc{z}=2$.

Implications and Theoretical Significance

The results establish that the voter model, despite lacking detailed balance and interface tension, exhibits universal scaling functions governed by Schrödinger-invariance. The explicit analytical forms for the scaling functions in arbitrary dimension provide a benchmark for testing dynamical symmetries in non-equilibrium systems. The identification of the upper critical dimension $d^*=2$ and the distinct scaling regimes for $d<2$ and $d>2$ clarify the crossover between fluctuation-dominated and mean-field behavior.

The work also highlights the role of composite operators in determining universal properties and demonstrates the irrelevance of non-linearities in the long-time limit for the voter model. The methodology developed can be extended to other models with $\mathpzc{z}=2$, including quantum systems and systems with more complex noise structures.

Future Directions and Relevance to AI

The analytical framework and symmetry-based approach presented in this work have potential applications in the paper of non-equilibrium dynamics in complex systems, including those relevant to AI and machine learning. For example, understanding the scaling and relaxation properties of high-dimensional stochastic systems can inform the design and analysis of algorithms for distributed consensus, opinion dynamics, and networked learning. The connection between dynamical symmetries and universal scaling functions may also inspire new approaches to model reduction and coarse-graining in large-scale AI systems.

Conclusion

This paper provides a rigorous and detailed analysis of the voter model, establishing the emergence of Schrödinger-invariance in its ageing dynamics across all spatial dimensions. The exact results for correlation and response functions, their scaling forms, and the identification of the relevant dynamical symmetries represent a significant advance in the understanding of non-equilibrium critical dynamics. The findings have broad implications for statistical mechanics, field theory, and the paper of complex systems, and open avenues for further exploration of dynamical symmetries in both classical and quantum non-equilibrium models.