arXiv subjects

Luca Makowiec

Publications and source records attributed to Luca Makowiec.

7 recordsLinked to original sources

arXiv2026

Fixed forests in the minimum spanning tree and cubic volume growth

Let $M_n$ be the minimum spanning tree of the complete graph $K_n$ with i.i.d.\ uniform edge weights. For a fixed forest $F$ with connected components $T_1, \ldots, T_d$, we show that there exists a function $Ψ$ on finite trees such that $$ n^{|E(F)|} \mathbb{P}_n(F \subseteq M_n) \longrightarrow \prod_{i=1}^d Ψ(T_i). $$ We give a recursive description of $Ψ$ and calculate it explicitly for several small trees. For the star $S_k$ and the path $P_k$, we prove that $Ψ(S_k) \sim ζ(2)^k$ and $Ψ(P_k) \sim k^2/12$, respectively. We also show that the expected size of a ball of radius $r$ is asymptotic to $r^3/36$, and give exponential tail bounds.

math.PR↗

arXiv2026

Critical curve of loop percolation on the $d$-regular tree

We consider clusters formed by a Poisson ensemble of random walk loops on the $d$-regular tree with an intensity parameter $α>0$ and a killing parameter $κ>-1$; the latter penalizes ($κ> 0$) or favors ($κ<0$) the appearance of large loops. We obtain an implicit formula for the critical curve $κ\mapsto α_c(κ)$ for the percolation phase transition; the curve is positive if and only if $κ>κ_c = \frac{2\sqrt{d-1}}{d}-1$, differentiable away from $κ_c$, and has order $\sqrt{κ-κ_c}$ as $κ\downarrowκ_c$ and order $(1+κ)^2$ as $κ\to\infty$. We show that for each $κ>-1$, an infinite cluster exists exactly when $α>α_c(κ)$. Finally, we identify the near-critical behavior of the susceptibility and the percolation probability: for $κ>κ_c$, the critical exponents take the mean-field values, while for $κ=κ_c$, the phase transition is of a higher order with the percolation probability decaying quadratically in $α-α_c$.

math.PR↗

arXiv2026

Random spanning trees in random environment

We introduce a new spanning tree model called the random spanning tree in random environment (RSTRE), which interpolates between the uniform spanning tree and the minimum spanning tree as the inverse temperature (disorder strength) $β$ varies. On the complete graph with $n$ vertices and i.i.d.\ uniform disorder variables on the edges, we identify: (1) a low disorder regime with $β\leq C n/\log n$, where the diameter of the random spanning tree is typically of order $n^{1/2}$, the same as for the uniform spanning tree; (2) a high disorder regime with $β\geq n^{4/3} \log n$, where the diameter is typically of order $n^{1/3}$, the same as for the minimum spanning tree. We conjecture that for $β=n^α$ with $α\in (1, 4/3)$, the diameter is of order $n^{γ+o(1)}$ for some $γ=γ(α)$ strictly between $1/2$ and $1/3$.

math.PR↗

arXiv2026

Local limits of random spanning trees in random environment

We study the edge overlap and local limit of the random spanning tree in random environment (RSTRE) on the complete graph with $n$ vertices and weights given by $\exp(-βω_e)$ for $ω_e$ uniformly distributed on $[0,1]$. We show that for $β$ growing with $β= o(n/\log n)$, the edge overlap is $(1+o(1)) β$, while for $β$ much larger than $n \log^2 n$, the edge overlap is $(1-o(1))n$. Furthermore, there is a transition of the local limit around $β= n$. When $β= o(n/ \log n)$ the RSTRE locally converges to the same limit as the uniform spanning tree, whereas for $β$ larger than $n \log^λn$, where $λ= λ(n) \rightarrow \infty$ arbitrarily slowly, the local limit of the RSTRE is the same as that of the minimum spanning tree.

math.PR↗

arXiv2025

Repeat times and a two-weight UST model

We study a model of random weighted uniform spanning trees on the complete graph with $n$ vertices, where each edge is assigned a weight of $n^{1+γ}$ with probability $1/n$ and $1$ otherwise. Whenever $γ$ is large enough, we prove that the diameter of the resulting tree is typically of order $n^{1/3} \log n$, up to a $\log \log n$ correction. Our approach uses estimates on repeat times for selecting components in a critical Erdős-Rényi graph, as well as concentration bounds on the sums of diameters of these components.

math.PR↗

arXiv2025

Observables of random spanning trees in random environment

In this thesis, we study a new disordered system called random spanning tree in random environment (RSTRE) across different families of graphs with varying disorder distributions. We examine several observables as functions of the disorder strength (inverse temperature) $β\geq 0$, and compare their values to the extreme cases $β= 0$ and $β\rightarrow \infty$, which correspond to the uniform spanning tree (UST) and the minimum spanning tree (MST), respectively. The results concerning the diameter are in line with those of arXiv:2311.01808 and arXiv:2410.16830, while the findings on local observables are based on arXiv:2410.16836. This thesis also includes new material on the RSTRE in the Euclidean infinite lattice, as well as a novel result on the diameter of the unweighted UST on a slightly supercritical random graph.

math.PR↗

arXiv2024

Diameter of uniform spanning trees on random weighted graphs

For any edge weight distribution, we consider the uniform spanning tree (UST) on finite graphs with i.i.d. random edge weights. We show that, for bounded degree expander graphs and finite boxes of ${\mathbb Z}^d$, the diameter of the UST is of order $n^{1/2+o(1)}$ with high probability, where $n$ is the number of vertices.

math.PR↗