Small-world networks are constructed by taking a regular lattice in d dimensions and randomly rewiring or adding a small fraction of long-range edges. The random walk mixing time, which measures how long the walk takes to spread nearly uniformly across all nodes, is known to drop dramatically somewhere between very sparse and moderate rewiring regimes. The precise question is whether this drop is a sharp threshold phenomenon, meaning the mixing time transitions abruptly from polynomial to polylogarithmic in the network size at a specific critical rewiring probability, or whether the transition is gradual and lacks a true critical point in dimensions three and above.