We address the fate of many-body localization (MBL) of mid-spectrum eigenstates of a matter-free $\mathrm{U}\left(1\right)$ quantum-link gauge theory Hamiltonian with random couplings on ladder geometries. Apart from level spacing distribution indicators like disorder-averaged mean level spacing, we also consider an intensive estimator $\mathcal{D}\in [0,1/4]$, which acts as a measure of elementary plaquettes on the lattice that are active or inert in mid-spectrum eigenstates as well as the concentration of these eigenstates in Fock space, with $\mathcal{D}$ equal to its maximum value of $1/4$ for Fock states in the electric flux basis. We calculate its distribution, $p\left(\mathcal{D}\right)$, for $L_x\times L_y$ lattices, with $L_y=2$ and 4, as a function of (a dimensionless) disorder strength $\alpha$ ($\alpha =0$ implies zero disorder) using exact diagonalization in many disorder realizations. Although finite-size estimators based on level spacings do not give a reliable critical disorder strength, ${\alpha }_c\left(L_y\right)$, beyond which MBL prevails as $L_x\to \infty$; a different estimator based on the skewness of $p\left(\mathcal{D}\right)$ gives ${\alpha }_c(L_y=2)=31.04\pm 0.54$ using data for $L_x\le 14$ due to faster convergence. $p\left(\mathcal{D}\right)$ for wider ladders with $L_y=4$ show a lower tendency to localize, suggesting a lack of MBL in two dimensions. A remarkable observation is the resolution of the (monotonic) infinite-temperature autocorrelation function of single plaquette diagonal operators in typical high-energy Fock states into a plethora of emergent timescales of increasing spatio-temporal heterogeneity as the disorder is increased. At intermediate $\alpha$ as well as for $\alpha$ slightly below ${\alpha }_c\left(L_y\right)$, a fraction of randomly selected initial Fock states display striking oscillatory temporal behavior of such plaquette operators in spatial regions formed out of connected plaquettes.

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• Eigenstate thermalization
• Many-body localization
• Quantum memories
• Quantum statistical mechanics
• Exact diagonalization
• Kinetically constrained models
• Lattice gauge theory