Non-uniqueness times for the maximizer of the KPZ fixed point.(English)Zbl 07826171
Summary: Let \(\mathfrak{h}_t\) be the KPZ fixed point started from any initial condition that guarantees \(\mathfrak{h}_t\) has a maximum at every time \(t\) almost surely. For any fixed \(t\), almost surely \(\max \mathfrak{h}_t\) is uniquely attained. However, there are exceptional times \(t \in (0, \infty)\) when \(\max \mathfrak{h}_t\) is achieved at multiple points. Let \(\mathcal{T}_k \subset (0, \infty)\) denote the set of times when \(\max \mathfrak{h}_t\) is achieved at exactly \(k\) points. We show that almost surely \(\mathcal{T}_2\) has Hausdorff dimension 2/3 and is dense, \(\mathcal{T}_3\) has Hausdorff dimension 1/3 and is dense, \(\mathcal{T}_4\) has Hausdorff dimension 0, and there are no times when \(\max \mathfrak{h}_t\) is achieved at 5 or more points. This resolves two conjectures of Corwin, Hammond, Hegde, and Matetski.
MSC:
| 82Cxx | Time-dependent statistical mechanics (dynamic and nonequilibrium) |
| 60Kxx | Special processes |
| 82Bxx | Equilibrium statistical mechanics |
Keywords:
KPZ fixed point;directed landscape;KPZ universality;Hausdorff dimension;Airy process;Airy sheetCite
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