Fluid Antenna Systems: A Geometric Approach to Error Probability and Fundamental Limits

Fluid antenna systems (FAS) utilize position reconfigurability to improve spatial diversity in wireless communications. However, a rigorous framework for error probability analysis under spatially correlated channels remains absent. This paper fills this gap by deriving a closed-form asymptotic expression for the symbol error rate (SER). This mathematical expression establishes the fundamental scaling law between the error performance and the spatial correlation matrix. A key insight from our analysis is that the achievable diversity gain depends entirely on the effective rank of the spatial channel, rather than the total number of antenna ports. To quantify this effective rank, we propose a dual approach: a theoretical derivation and a geometry-based algorithm. Both methods rigorously prove that the effective rank converges to a fundamental limit of 2W+1, where W denotes the normalized aperture width. Specifically, the geometry-based algorithm extracts distinct performance thresholds from the eigenvalue spectrum of the channel. These thresholds perfectly match the derived theoretical limit. Furthermore, the proposed effective rank model demonstrates higher accuracy than existing approaches in the literature. Based on this robust framework, we offer a complete characterization of diversity gains and coding gains. The analytical results reveal a definitive design principle: enlarging the physical aperture increases the effective rank and drives performance improvements, whereas simply increasing port density within a fixed aperture yields diminishing returns.