Committing to Bubbles: Finding the Critical Configuration on the Lattice

Abstract

The nucleation of bubbles in first-order phase transitions is traditionally characterised by the critical bubble: defined as the saddle-point solution of the Euclidean action that separates collapsing from expanding field configurations. While this picture is exact in the noiseless, zero-temperature limit, thermal fluctuations introduces stochasticity which can influence the behaviour of the field configuration. In this work, we develop a purely statistical criterion for identifying the critical bubble by leveraging the concept of the ``committor'' probability: the likelihood that a given local field configuration evolves to the true vacuum before returning to the false vacuum. Using ensembles of lattice simulations with controlled thermal noise, we extract the committor probability during the evolution of a bubble from sub- to super-criticality. We find this approach to be robust, accounts for finite-temperature effects, and allows independent verification of bounce-based predictions. To demonstrate this, we compare the average profile obtained via the committor probability method to standard theory for a given model and find strong agreement, particularly at the core of the bubble. Importantly, we also observe that the behaviour of the committor probability with time is smooth and well defined. This method establishes a robust, simulation-driven framework for studying nucleation dynamics in thermal field theories and may be especially applicable in cases where analytical control might be limited.

Figures (8)

• Figure 1: Example evolution for scalar potential considered in this paper when coupled to a thermal bath, shown in \ref{['tab:c_H']}. (left) rapid thermalisation around the metastable origin starting from $\phi(\mathbf{x},0)=0=\dot\phi(\mathbf{x},0)$. (right) on a much longer timescale the system transitions to the true vacuum. The red dashed line marks $\phi_{\rm vev}$ of the continuum effective theory which agrees with the lattice simulation due to the inclusion of lattice renormalisation counterterms, such as $Z_\phi$, $\delta \lambda$ and most importantly $\delta m^2$.
• Figure 2: (top) An illustration of the initial simulation procedure. A large lattice volume is simulated until a region of the lattice reaches the true-vacuum position, in this case indicated by blue. Once a small region has formed, this corresponds to a phase transition where this physical region developed a field profile which moved from sub- to super-criticality in time. This data is stored and then used as initial conditions on future, much smaller simulations, to determine at which exact time slice the criticality occurred. (bottom) The bottom plot shows that if the simulation was allowed to continue this region would develop into an expanding bubble including the nucleation of more bubbles at other lattice sites (here shown in red). The simulation procedure closesly follows what was preformed in Dutka:2025oqt.
• Figure 3: Dependence of the survival probability: the probability of an ensemble of simulations to have nucleated a bubble by time $t$ with different lattice volumes, in units of $T$. The theoretical expectation is $\ln P_{\rm surv}(t)/V = \mathrm{const} - \Gamma t$. For small lattice volumes (left) the expected behaviour is not recovered whereas for sufficiently large lattice volumes (right), the survival probabilities line up. For comparison, semiclassical predictions based on the bounce action are also shown. The curve labeled “BubbleDet” corresponds to the full nucleation rate $\Gamma = A_{\rm BD} \exp(-S_3/T)$, where the prefactor $A_{\rm BD}$ is computed numerically using BubbleDetEkstedt:2023sqc. The curve labeled “bounce only” shows the exponential suppression factor $\exp(-S_3/T)$ only, which is obtained from CosmoTransitionsWainwright:2011kj. This comparison illustrates the importance of the prefactor contribution to the nucleation rate in this benchmark This suggests a minimum lattice volume for the potential in \ref{['tab:c_H']} when performing the secondary simulations to determine $p_{\rm B}(\phi_0(t))$ in order to have consistent behaviour.
• Figure 4: Example evolution of the bubble core, $\langle|\phi_{\rm core}|\rangle$, in the secondary simulations. Blue lines indicate simulations where the core grows toward $\phi_{\rm TV}$, while red lines correspond to cores remaining near the metastable vacuum. The top row shows simulations with initial conditions $(\phi,\pi)$ taken from the primary simulation around a bubble evolving from sub- to super-critical, whereas the bottom row shows control simulations without profile injection. Panels from left to right correspond to increasingly late times for the initial profiles $\phi_0(t)$ and $\pi_0(t)$. As the initial profiles become more critical, more cores evolve to the true vacuum. All control simulations remain near the metastable vacuum; since the two sets share common random numbers (CNR), the observed growth in the top row is entirely due to the injected profiles.
• Figure 5: Example plots of the committor probability $p_{\rm B}(\phi_0(t))$ for different primary simulations. In most cases, $p_{\rm B}$ evolves monotonically with time, reflecting the smooth growth of the injected profile toward criticality. The transition from $p_{\rm B} \approx 0$ to $p_{\rm B} \approx 1$ identifies the formation of the critical bubble in each simulation.