Personalized and continuous interactions are critical for LLM-based conversational agents, yet finite context windows and static parametric memory hinder the modeling of long-term, cross-session user states. Existing approaches, including retrieval-augmented generation and explicit memory systems, primarily operate at the fact level, making it difficult to distill stable preferences and deep user traits from evolving and potentially conflicting dialogues.To address this challenge, we propose RGMem, a self-evolving memory framework inspired by the renormalization group (RG) perspective on multi-scale organization and emergence. RGMem models long-term conversational memory as a multi-scale evolutionary process: episodic interactions are transformed into semantic facts and user insights, which are then progressively integrated through hierarchical coarse-graining, thresholded updates, and rescaling into a dynamically evolving user profile.By explicitly separating fast-changing evidence from slow-varying traits and enabling non-linear, phase-transition-like dynamics, RGMem enables robust personalization beyond flat retrieval or static summarization. Extensive experiments on the LOCOMO and PersonaMem benchmarks demonstrate that RGMem consistently outperforms SOTA memory systems, achieving stronger cross-session continuity and improved adaptation to evolving user preferences. Code is available at https://github.com/fenhg297/RGMem