  We establish the first connection between $2d$ Liouville quantum gravity and
natural dynamics of random matrices. In particular, we show that if $(U_t)$ is
a Brownian motion on the unitary group at equilibrium, then the measures $$
|\det(U_t - e^{i \theta})|^{\gamma} dt d\theta $$ converge in the limit of
large dimension to the $2d$ LQG measure, a properly normalized exponential of
the $2d$ Gaussian free field. Gaussian free field type fluctuations associated
with these dynamics were first established by Spohn (1998) and convergence to
the LQG measure in $2d$ settings was conjectured since the work of Webb (2014),
who proved the convergence of related one dimensional measures by using inputs
from Riemann-Hilbert theory.
  The convergence follows from the first multi-time extension of the result by
Widom (1973) on Fisher-Hartwig asymptotics of Toeplitz determinants with real
symbols. To prove these, we develop a general surgery argument and combine
determinantal point processes estimates with stochastic analysis on Lie group,
providing in passing a probabilistic proof of Webb's $1d$ result. We believe
the techniques will be more broadly applicable to matrix dynamics out of
equilibrium, joint moments of determinants for classes of correlated random
matrices, and the characteristic polynomial of non-Hermitian random matrices.
