Dynamical Large Deviations of Two-Dimensional Kinetically Constrained Models Using a Neural-Network State Ansatz

We use a neural-network ansatz originally designed for the variational optimization of quantum systems to study dynamical large deviations in classical ones. We use recurrent neural networks to describe the large deviations of the dynamical activity of model glasses, kinetically constrained models in two dimensions. We present the first finite size-scaling analysis of the large-deviation functions of the two-dimensional Fredrickson-Andersen model, and explore the spatial structure of the high-activity sector of the South-or-East model. These results provide a new route to the study of dynamical large-deviation functions, and highlight the broad applicability of the neural-network state ansatz across domains in physics.

Figure 1

(a) For a two-dimensional system, an RNN cell with learnable parameters $\mathcal{W}$ calculates a new hidden state $h$, given the hidden and visible states of previous lattice sites $\alpha$ and $\beta$, and passes this state to other lattice sites. The hidden state is further processed to calculate the normalized conditional probability amplitude $\psi (x_i|\{x_{j<i}\})$. (b) This RNN cell is applied to each site of a two-dimensional lattice, and the RNN traverses the lattice row by row in a zigzag path to calculate the total probability amplitude of a configuration. Note that the hidden state is passed on in both the vertical and horizontal direction, respecting the geometry of the system under study. This probability amplitude is then used for the variational optimization of the scaled cumulant-generating function $\theta (s)$.

Figure 2

(a) Scaled cumulant-generating function $\theta (s)$ of the two-dimensional Fredrickson-Andersen model at $c=0.5$ with two-dimensional recurrent neural-network states, on $L\times L$ square lattices with length between $L=8$ and $L=32$. Inset: scaling of the location $s_c$ of the singularity in the SCGF with the number of lattice sites for three values of $c$. (b) The dynamical activity ${⟨k⟩}_s=-{\theta }^{\prime }(s)$ per lattice site. (c) The rate function $J(k)$ for $L=8$, 16, 32.

Figure 3

Average density in the active sector ($s<0$) of the South-or-East model, as a function of $\nu \equiv 1-e^s$, for an $8\times 8$ system at $c=0.1$. On the right, we show a density profile for each of the levels in $⟨n{⟩}_{\nu }$.