Thick points of 4D critical branching Brownian motion

TL;DR

This work uncovers a rich multifractal geometry for thick points of critical branching Brownian motion in four dimensions, providing exact tail exponents, a double-phase transition, and a dimension formula for thick-point sets. It develops a comprehensive framework built on BGW trees, tree-indexed processes, and PDE methods for nonpositive solutions of Δv=v^2, yielding probabilistic representations, uniqueness results, and boundary-value analyses. A key contribution is the sharp, dimension-dependent asymptotics for hitting probabilities and the thick-point counts, together with a strong coupling to branching random walk that transfers results to lattice models. Across dimensions, the authors reveal infinite-order asymptotics for hitting probabilities, noting a divergent expansion precisely at d=4, and derive universal exponents β(d) for d≠4. The handbook-style structure of the paper combines probabilistic techniques with PDE analysis and multiplicative-chaos ideas to illuminate both continuous and discrete branching systems and their occupation measures.

Abstract

We study the thick points of branching Brownian motion and branching random walk with a critical branching mechanism, focusing on the critical dimension $d = 4$. We determine the exponent governing the probability to hit a small ball with an exceptionally high number of pioneers, showing that this has a second-order transition between an exponential phase and a stretched-exponential phase at an explicit value ($a = 2$) of the thickness parameter $a$. We apply the outputs of this analysis to prove that the associated set of thick points $\mathcal{T}(a)$ has dimension $(4-a)_+$, so that there is a change in behaviour at $a=4$ but not at $a = 2$ in this case. Along the way, we obtain related results for the nonpositive solutions of a boundary value problem associated to the semilinear PDE $Δv = v^2$ and develop a strong coupling between tree-indexed random walk and tree-indexed Brownian motion that allows us to deduce analogues of some of our results in the discrete case. We also obtain in each dimension $d\geq 1$ an infinite-order asymptotic expansion for the probability that critical branching Brownian motion hits a distant unit ball, finding that this expansion is convergent when $d\neq 4$ and divergent when $d=4$. This reveals a novel, dimension-dependent critical exponent governing the higher-order terms of the expansion, which we compute in every dimension.

Figures (8)

• Figure 1.1: Plot of $\psi$ in blue and of the identity function in dotted orange. The function $\psi$ is differentiable but not twice differentiable at $a=2$.
• Figure 3.1: Plots of numerical approximations of $g_\lambda(x)$ for different positive values of $\lambda$. Left: If $\lambda$ is small enough (e.g. $\lambda<1$), $g_\lambda$ possesses a probabilistic representation (Theorem \ref{['T:proba_representation']}), and the curves for different small values of $\lambda$ cannot cross each other. Right: The numerical solutions plotted here suggest that this non-intersection property fails for larger values of $\lambda$, even when the solutions are non-positive. Further numerics suggest that the critical value for solutions to be monotonic in $\lambda$ is around $2.4$, see Figure \ref{['F:turning_point']}.
• Figure 3.2: Left: Numerical plots of $u_\lambda(t):=\lambda^{-1} g_\lambda(\lambda^{-1} t)$ for various large values of $\lambda$. These numerical computations suggest that $g_\lambda$can become positive when $\lambda$ is large. The solutions with $\lambda=150$ and $\lambda=500$ cross the $x$-axis outside of the plot. Further numerics suggest that the critical value of $\lambda$ for $g_\lambda$ to remain non-positive is roughly $11.2$
• Figure 3.3: Numerical plots of $-\frac{x}{2}g_\lambda(x)$ with $x=1000$ as a function of $\lambda$. Left: The asymptotics $g_\lambda(x)\sim -2/x$ as $x\to \infty$, proven for fixed $\lambda \in (0,1)$ in Section \ref{['sec:series_expansion']}, is verified numerically for a larger range of $\lambda$ values. The convergence appears to hold uniformly away from $0$ and some critical value around $11.25$. Right: Zooming in on this plot near its maximum. The maximum of the function is attained numerically at $\lambda \approx 2.43$, suggesting that the critical value for our probabilistic representation of solutions to hold is around $\inf _R \lambda_c(R) \approx \lambda_c(e^{1000})\approx 2.43$.
• Figure 4.1: Left: Numerical plots of $u_\lambda(t):=\lambda^{-1} g_\lambda(\lambda^{-1} t)$ for various values of $0\leq \lambda \leq 2$. The first two phases become near-instantaneous under this scaling as $\lambda \downarrow 0$. Right: Numerical plots of $u_\lambda(t):=\lambda^{-1} g_\lambda(\lambda^{-1}t)$ (blue) and its first three derivatives (yellow, green, and red respectively) with $\lambda = 1/8$. The values $t=\lambda x_0$ and $t=\lambda x_1$ corresponding to $x_0$ and $x_1$ are represented by dashed lines. Here we have zoomed in on smaller values of $x$ to give more insight into the first two phases.

Theorems & Definitions (125)

• Remark 1.1
• Theorem 1.2: Exponents for the number of pioneers
• Theorem 1.3
• Theorem 1.4: Exponents for the local time
• Theorem 1.5
• Remark 1.6
• Theorem 1.7: Hitting probabilities in the critical dimension
• Remark 1.8
• Theorem 1.9: Hitting probabilities in noncritical dimensions
• Theorem 1.10