2026 Theses Doctoral

Heterogeneous Interactions in Particle Systems, Langevin Sampling, and Stochastic Games

Zhou, Fuzhong

Large systems of interacting components are often studied under homogeneity assumptions, which allow powerful probabilistic techniques to reduce high-dimensional problems to tractable mean-field limits. In the presence of heterogeneous interactions, however, these classical approaches break down, and many standard tools no longer apply. Fundamental questions concerning marginal structures and limiting behavior consequently become more delicate, and developing analytical frameworks that remain effective without exchangeability emerges as a central challenge. This thesis addresses several aspects of this challenge, with a particular focus on extending classical mean-field theory to high-dimensional probabilistic models with heterogeneous interaction structures.

The first part of the thesis studies interacting diffusive particle systems with nonsymmetric pairwise interaction strengths, where classical propagation of chaos results, which rely heavily on exchangeability, fail to apply. A non-asymptotic framework for mean-field approximations in such non-exchangeable systems is developed. In particular, the marginal law of any subset of particles is compared with a suitably chosen product measure, yielding sharp bounds on the corresponding relative entropy. Extending earlier work in the exchangeable setting, we employ a generalized BBGKY hierarchy to derive a system of differential inequalities governing marginal relative entropies. A key step is an unexpected connection with first-passage percolation, which enables control of these marginal entropies via expectations of functionals of an associated percolation process.

The second part of the thesis applies related ideas to the analysis of high-dimensional sampling algorithms. The unadjusted Langevin algorithm is widely used for sampling from complex distributions but is known to exhibit bias that typically grows with the ambient dimension. Recent work identified a delocalization phenomenon in sparse interaction models, showing that the bias of low-dimensional marginals depends only on the marginal dimension, rather than the full system dimension. Building on this insight, we strengthen these results by removing a logarithmic factor in the sparse interaction regime, measuring bias in relative entropy, and relaxing the strong log-concavity assumption. In addition, the delocalization phenomenon is extended to distributions with weak interactions. The analysis relies on a hierarchical study of marginal relative entropies, drawing methodological connections to the propagation of chaos framework developed for heterogeneous particle systems.

The final part turns to strategic interactions among large populations of agents. Graphon games generalize mean-field games by allowing heterogeneous interaction patterns across players, thereby providing a flexible framework for large-scale games with network structure. We introduce a general discrete-time formulation of graphon games based on a representative player. This complements recent developments in continuous time and provides a self-contained framework that facilitates algorithmic analysis. Under mild assumptions, equilibrium properties are rigorously established. Building upon this analytical foundation, we develop an approximate fixed-point iteration framework that leads to an oracle-free online algorithm for computing equilibria, together with corresponding sample complexity guarantees.