High-energy hadron colliders are designed to generate particle collisions within specialized detectors. A higher number of collisions is achieved with high-quality beams of low transverse emittances, meaning a small transverse cross-section, and high intensity, meaning many particles per bunch. This thesis studies how noise negatively impacts the beam quality in high-energy hadron colliders, both in terms of beam instabilities and emittance growth. The impact is analyzed through the derivation of new theories, multi-particle tracking simulations, the numerical solving of partial differential equations, and dedicated experiments in CERN's Large Hadron Collider (LHC).

The impact of noise on beam stability cannot be treated with the first-order, linear Vlasov equation, which is commonly used to study the thresholds of collective instabilities. Therefore, the Vlasov equation has in this thesis been expanded to second order in the perturbation of the beam distribution, finding a diffusion mechanism driven by the interplay between noise, decoherence, and wakefields. The diffusion leads to a local flattening of the distribution, which can cause a loss of Landau damping after a time delay referred to as the latency. An analytical formula for the latency and a specialized numerical diffusion solver were successfully benchmarked against the latency measurements in a dedicated experiment conducted in the LHC. Precaution in the machine operation has to be taken to account for this new mechanism. In particular, it is found that the machine must be operated with a margin to the linear stability threshold. For the case of the LHC, it has previously been found empirically that the octupole current during operation must be increased by about a factor 2, and this thesis provides the explanation as to why that is. Alternative operational settings are suggested to reduce the required octupole current in the LHC. In addition, the new theory allows for extrapolations to future machines, such as the High-Luminosity LHC, as well as the estimation of the impact of new devices, such as crab cavities.

External noise and noise from the transverse beam feedback system cause an emittance growth rate due to decoherence of the noise kicks. Analytical theories for the suppression of the emittance growth rate with a bunch-by-bunch feedback have here been extended to a multi-bunch feedback. The numerical study of suppression during collision was conducted by means of a newly developed parallel multi-beam multi-bunch algorithm. For the typical case of low-frequency external noise and non-negligible feedback noise, a multi-bunch feedback has both analytically and numerically been found superior to a bunch-by-bunch feedback, as it can suppress the impact of the external noise equally well, while simultaneously reducing the noise generated by the feedback itself. Suggestions for a more optimal operation of the LHC are discussed, including a reduction of the upper cutoff frequency of the feedback system.