A branch-stationary max-stable field on a binary tree

Vertices are words over the alphabet {L, R}. Along each edge an independent Gaussian increment with variance Δ is added, giving a branching random walk W. The field is η_v = max_i P_i exp(W_v⁽ⁱ⁾ − Δ|v|/2), with P_i the points of a Poisson process with intensity p⁻²dp. In the infinite construction every subtree has the same finite-dimensional laws as the whole tree — that is branch-stationarity. The picture uses a finite truncation of that construction. From Hashorva & Novikov (2026).

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Edge variance Δ = 1.0 Depth = 6 Poisson points retained = 40

Small Δ: the same Poisson point tends to win everywhere (one colour). Large Δ: the winning point changes from branch to branch — the colours form smaller patches as the branching random walk spreads out. This does not assert independence. With finitely many Poisson points the maxima are underestimated, especially far from the root when Δ is large; this approximation is not itself branch-stationary. Increase the retained count to reduce truncation. Value colours are scaled within the displayed subtree.