Fuzzing is highly effective in detecting bugs due to the key contribution of randomness. However, randomness significantly reduces the efficiency of fuzzing, causing it to cost days or weeks to expose bugs. Even though directed fuzzing reduces randomness by guiding fuzzing towards target buggy locations, the dilemma of randomness still challenges directed fuzzers. Two critical components, which are seeds and mutators, contain randomness and are closely tied to the conditions required for triggering bugs. Therefore, to address the challenge of randomness, we propose to use large language models (LLMs) to remove the randomness in seeds and reduce the randomness in mutators. With their strong reasoning and code generation capabilities, LLMs can be used to generate reachable seeds that target pre-determined locations and to construct bug-specific mutators tailored for specific bugs. We propose RandLuzz, which integrates LLMs and directed fuzzing, to improve the quality of seeds and mutators, resulting in efficient bug exposure. RandLuzz analyzes function call chain or functionality to guide LLMs in generating reachable seeds. To construct bug-specific mutators, RandLuzz uses LLMs to perform bug analysis, obtaining information such as bug causes and mutation suggestions, which further help generate code that performs bug-specific mutations. We evaluate RandLuzz by comparing it with four state-of-the-art directed fuzzers, AFLGo, Beacon, WindRanger, and SelectFuzz. With RandLuzz-generated seeds, the fuzzers achieve an average speedup ranging from 2.1$\times$ to 4.8$\times$ compared to using widely-used initial seeds. Additionally, when evaluated on individual bugs, RandLuzz achieves up to a 2.7$\times$ speedup compared to the second-fastest exposure. On 8 bugs, RandLuzz can even expose them within 60 seconds.