Committing to Bubbles: Finding the Critical Configuration on the Lattice

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.