Mathematics Seminar Series: Probability

Title:  Inhomogeneous Long-Range First-Passage Percolation under random vertex environment.

Abstract:  We introduce an inhomogeneous generalization of the long-range first-passage percolation model. This model combines long-range geometry with a heavy-tailed vertex environment and general positive edge noise. It may be viewed as an inhomogeneous, vertex-weighted extension of classical long-range first-passage percolation. It can also be seen as a dense analog of sparse spatial-scale-free first-passage models. A distinctive feature of the model is that its large-scale metric behavior is governed by three competing mechanisms:

1. hub effects created by exceptionally large vertex weights, controlled by the vertex-tail exponent $\gamma$,
2. fast-edge effects created by exceptionally small edge noises, controlled by the edge-noise lower-tail exponent $\theta$, and
3. long-edge jumps, controlled by the distance-cost exponent $\alpha$.

For the d-dimensional square lattice, this creates a rich phase diagram. Regimes of instantaneous, tight, stretched-exponential, superlinear, and linear growth appear, depending on the three exponents $\alpha,\gamma,\theta$. I will explain the mechanisms behind these phase transitions. I will also highlight how the resulting metric behavior differs from both classical long-range first-passage percolation and sparse scale-free first-passage models. Joint work with Shirshendu Chatterjee and Partha Dey.