Static analysis is a widely used technique in software engineering for identifying and mitigating bugs. However, a significant hurdle lies in achieving a delicate balance between precision and scalability. Large Language Models (LLMs) offer a promising alternative, as recent advances demonstrate remarkable capabilities in comprehending, generating, and even debugging code. Yet, the logic of bugs can be complex and require sophisticated reasoning and a large analysis scope spanning multiple functions. Therefore, at this point, LLMs are better used in an assistive role to complement static analysis. In this paper, we take a deep dive into the open space of LLM-assisted static analysis, using use-before-initialization (UBI) bugs as a case study. To this end, we develop LLift, a fully automated framework that interfaces with both a static analysis tool and an LLM. By carefully designing the framework and the prompts, we are able to overcome a number of challenges, including bug-specific modeling, the large problem scope, the non-deterministic nature of LLMs, etc. Tested in a real-world scenario analyzing nearly a thousand potential UBI bugs produced by static analysis, LLift demonstrates a potent capability, showcasing a reasonable precision (50%) and appearing to have no missing bugs. It even identified 13 previously unknown UBI bugs in the Linux kernel. This research paves the way for new opportunities and methodologies in using LLMs for bug discovery in extensive, real-world datasets.