AC Magnetic Susceptibility and $\mu $SR Study of Spin Dynamics and the Onset of Magnetic Correlations in LiHo$_{x}$Y$_{1-x}$F$_{4}$

The onset of correlation effects in the magnetic Ho$^{3+}$-subsystem in LiHo$_{x}$Y$_{1-x}$F$_{4}$ single crystals is studied by comparing measurements and simulations of the field and frequency dependent magnetic AC susceptibility at 1.8 K and field and temperature dependent muon depolarization rates ($\mu $SR) for the concentrations x=0.0017, 0.0085, 0.041 and 0.0855. Specific features in the field and frequency dependence of in-phase and out-of-phase susceptibilities, in particular, broadening of peaks (dips) in $\chi $' ($\chi $'') that indicate enhanced relaxation processes at field induced avoided level crossings, can be associated with x-dependent changes of cross relaxation rates and the phonon bottleneck effect in the spin-lattice relaxation. The observed peak in the measured temperature dependent muon relaxation rate appears to be related to a maximum at the frequency 60 cm$^{-1}$ in the acoustic phonon density of states.