R. Arratia (The Motion of a Tagged Particle in the Simple Symmetric Exclusion System on Z) shows in theorem 2 that for step initial condition in the SSEP, the position of the lead particle, $x_1(t)$ has a strong law of large numbers. $(\frac{x_1(t)}{\sqrt{t}} - \sqrt{\log(t)}) \rightarrow 0$ almost surely. Are there other models with such behavior?
2026-04-01 03:16:27.1775013387
Are there any models that have mean $\sqrt{t\log(t)}$?
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