Tom Hutchcroft

math.PR2026

Super-Brownian limits and the $k$ -point function for high-dimensional percolation

Arthur Blanc-Renaudie, Tom Hutchcroft

We prove that there exist positive constants $A$ and $V$ such that the high-dimensional critical percolation $k$ -point function is given by \[ T_{p_c}(x_1,x_2,\ldots,x_{k}) \sim V^…

math.PR2025

A discontinuous percolation phase transition on the hierarchical lattice

Johannes BÃ¤umler, Tom Hutchcroft

For long-range percolation on $Z$ with translation-invariant edge kernel $J$ , it is a classical theorem of Aizenman and Newman (1986) that the phase transition is disconti…

math.PR2025

Critical long-range percolation III: The upper critical dimension

Tom Hutchcroft

In long-range percolation on $Z^{d}$ , points $x$ and $y$ are connected by an edge with probability $1 - exp (- I^^{2} ∥ x - y ∥^{- d - I^\pm})$ , where $I^\pm > 0$ is fixed and $I^^{2} \geq 0$ is a pa…

math.PR2025

Critical long-range percolation II: Low effective dimension

Tom Hutchcroft

In long-range percolation on $Z^{d}$ , points $x$ and $y$ are connected by an edge with probability $1 - exp (- I^^{2} ∥ x - y ∥^{- d - I^\pm})$ , where $I^\pm > 0$ is fixed and $I^^{2} \geq 0$ is a pa…

math.PR2025

Critical long-range percolation I: High effective dimension

Tom Hutchcroft

In long-range percolation on $Z^{d}$ , points $x$ and $y$ are connected by an edge with probability $1 - exp (- I^^{2} ∥ x - y ∥^{- d - I^\pm})$ , where $I^\pm > 0$ is fixed and $I^^{2} \geq 0$ is a pa…

math.PR2024

Dimension jump at the uniqueness threshold for percolation in $\infty + d$ dimensions

Tom Hutchcroft, Minghao Pan

Consider percolation on $T \times Z^{d}$ , the product of a regular tree of degree $k \geq 3$ with the hypercubic lattice $Z^{d}$ . It is known that this graph has $0<p_c…

Super-Brownian limits and the k-point function for high-dimensional percolation

A discontinuous percolation phase transition on the hierarchical lattice

Critical long-range percolation III: The upper critical dimension

Critical long-range percolation II: Low effective dimension

Critical long-range percolation I: High effective dimension

Dimension jump at the uniqueness threshold for percolation in ∞+d dimensions

Super-Brownian limits and the $k$ -point function for high-dimensional percolation

Dimension jump at the uniqueness threshold for percolation in $\infty + d$ dimensions