Finite temperature dopant-induced spin reorganization explored via tensor networks in the two-dimensional $t$-$J$ model

We study the two-dimensional $t$--$J$ model at finite temperature directly in the thermodynamic limit using purification represented by an infinite projected entangled-pair state (iPEPS). We reach temperatures down to $T/t=0.1$ and hole concentrations up to $1-n\simeq0.25$, and provide benchmark thermodynamic-limit results for the specific heat, uniform susceptibility, and charge compressibility. We identify a susceptibility maximum $T^\ast$ that tracks the buildup of short-range antiferromagnetism and a shallow compressibility enhancement upon cooling in the same doping window. To expose the underlying microscopic mechanism, we introduce dopant-conditioned multi-point correlators that quantify how holes reorganize nearby exchange: single holes weaken adjacent antiferromagnetic bonds, while nearest-neighbor hole pairs produce a cooperative response that reinforces antiferromagnetism on the parallel plaquette edge. Over the same parameter window, $d$-wave pairing correlations remain short-ranged. These results provide experiment-compatible thermodynamic-limit benchmarks and establish dopant-conditioned correlators as incisive probes of finite-temperature spin-texture reorganization in doped Mott insulators.