arXiv subjects

William Da Silva

Publications and source records attributed to William Da Silva.

9 recordsLinked to original sources

arXiv2026

The self-dual point of Fortuin--Kasteleyn planar maps is critical

We study the Fortuin--Kasteleyn model of planar maps with parameter $q\in (0,4)$ at and away from its self-dual point. This model is also bijectively equivalent to the fully packed (bicoloured) loop-$O(n)$ model on planar triangulations. These have been traditionally studied using either techniques from analytic combinatorics (based in particular on the gasket decomposition of Borot, Bouttier and Guitter) or probabilistic arguments (based on Sheffield's hamburger-cheeseburger bijection). In this paper we establish a dictionary relating quantities of interest in both approaches. This has several consequences. First, we derive an exact expression for the partition function of the fully packed (colour-symmetric) loop-$O(n)$ model on triangulations, as a function of the outer boundary length. This confirms predictions by Gaudin and Kostov. In particular, this model exhibits critical behaviour, in the sense that the partition function exhibits a power-law decay characteristic of the critical regime at this self-dual point. Secondly, we derive precise asymptotics for geometric features of the self-dual Fortuin--Kasteleyn model of planar maps when $0 < q <4$, such as the exact polynomial tail behaviour of the perimeters of clusters and loops. This sharpens previous results of arXiv:1502.00450 and arXiv:1502.00546. Finally, we prove that Fortuin--Kasteleyn maps undergo a \emph{sharp} phase transition at the self-dual point in the sense that, away from the self-dual point, cluster sizes decay exponentially. This mirrors the celebrated result by Beffara and Duminil-Copin for $q\geq 1$ on a fixed lattice and constitutes the first result establishing that the self-dual point of Fortuin--Kasteleyn maps is the critical point. A key step of our proof is to use the above dictionary and the probabilistic results to justify rigorously an ansatz commonly assumed in the analytic combinatorics literature.

math.PR↗

arXiv2026

Scaling limits of critical FK-decorated random planar maps with $q=4$

We establish the first scaling limit for FK($q$)-weighted planar maps in the critical case $q=4$, resolving a problem that has remained open since Sheffield's seminal work arXiv:1108.2241. In that work, Sheffield proved a scaling limit for $q<4$ via the celebrated hamburger-cheeseburger bijection, which initiated the peanosphere (mating-of-trees) approach to Liouville quantum gravity. We prove that, at criticality, the associated burger count $\mathcal{S}$ and discrepancy $\mathcal{D}$ satisfy \[ \left(\frac{\mathcal{S}_{\lfloor nt \rfloor}}{\sqrt{n}}, \frac{\log(n)}{{2π}\sqrt{n}} \mathcal{D}_{\lfloor nt \rfloor}\right)_{t\in\mathbb{R}} \stackrel{\text{d}}{\longrightarrow} (B^1_t, B^2_{t})_{t\in\mathbb{R}}, \] where $B^1$ and $B^2$ are independent two-sided Brownian motions. To the best of our knowledge, no conjecture for the correct discrepancy scaling factor had previously been formulated. Matching the limiting process with the critical mating of trees arXiv:2109.00275, we establish the first rigorous planar map convergence towards CLE$_4$ and critical ($γ=2$) Liouville quantum gravity, in the peanosphere sense. Our proof is based on a novel approach that reveals the exactly solvable nature of the model through a correspondence with the (bicoloured) fully packed loop-$O(2)$ model on triangulations, and yields critical geometric exponents matching the predictions of conformal field theory.

math.PR↗

arXiv2025

The scaling limit of the volume of loop O(n) quadrangulations

We study the volume of rigid loop-$O(n)$ quadrangulations with a boundary of length $2p$ in the non-generic critical regime. We prove that, as the half-perimeter $p$ goes to infinity, the volume scales in distribution to an explicit random variable. This limiting random variable is described in terms of the multiplicative cascades of Chen, Curien and Maillard arXiv:1702.06916, or alternatively (in the dilute case) as the law of the area of a unit-boundary $γ$-quantum disc, as determined by Ang and Gwynne arXiv:1903.09120, for suitable $γ$. Our arguments go through a classification of the map into several regions, where we rule out the contribution of bad regions to be left with a tractable portion of the map. One key observable for this classification is a Markov chain which explores the nested loops around a size-biased vertex pick in the map, making explicit the spinal structure of the discrete multiplicative cascade. We stress that our techniques enable us to include the boundary case $n=2$, that we define rigorously, and where the nested cascade structure is that of a critical branching random walk. In that case the scaling limit is given by the limit of the derivative martingale and is inverse-exponentially distributed, which answers a conjecture of arXiv:2005.06372v2.

math.PR↗

arXiv2025

The longest increasing subsequence of Brownian separable permutons

We establish a scaling limit result for the length $\operatorname{LIS}(σ_n)$ of the longest increasing subsequence of a permutation $σ_n$ of size $n$ sampled from the Brownian separable permuton $\boldsymbolμ_p$ of parameter $p\in(0,1)$, which is the universal limit of pattern-avoiding permutations. Specifically, we prove that \[\frac{\operatorname{LIS}(σ_n)}{n^α}\;\underset{n\to\infty}{\overset{\mathrm{a.s.}}{\longrightarrow}}\; X,\] where $α=α(p)$ is the unique solution in the interval $(1/2,1)$ to the equation \[\frac{1}{4^{\frac{1}{2α}}\sqrtπ}\,\frac{Γ\big(\tfrac{1}{2}-\tfrac{1}{2α}\big)}{Γ\big(1-\tfrac{1}{2α}\big)}=\frac{p}{p-1},\] and $X=X(p)$ is a non-deterministic and a.s. positive and finite random variable, which is a measurable function of the Brownian separable permuton. Notably, the exponent $α(p)$ is an increasing continuous function of $p$ with $α(0^+)=1/2$, $α(1^-)=1$ and $α(1/2)\approx0.815226$, which corresponds to the permuton limit of uniform separable permutations. We prove analogous results for the size of the largest clique of a graph sampled from the Brownian cographon of parameter $p\in(0,1)$.

math.PR↗

arXiv2025

Growth-fragmentations, Brownian cone excursions and SLE(6) explorations of a quantum disc

The aim of this article is to present a growth-fragmentation process naturally embedded in a Brownian excursion from boundary to apex in a cone of angle $2π/3$. This growth-fragmentation process corresponds, via the so-called mating-of-trees encoding arXiv:1409.7055, to the quantum boundary length process associated with a branching $\mathrm{SLE}_6$ exploration of a $γ= \sqrt{8/3}$ quantum disc. However, our proof uses only Brownian motion techniques, and along the way we discover various properties of Brownian cone excursions and their connections with stable Lévy processes. Assuming the mating of trees encoding, our results imply several fundamental properties of the $γ= \sqrt{8/3}$-quantum disc $\mathrm{SLE}_6$-exploration.

math.PR↗

arXiv2024

Power-law bounds for increasing subsequences in Brownian separable permutons and homogeneous sets in Brownian cographons

The Brownian separable permutons are a one-parameter family -- indexed by $p\in(0,1)$ -- of universal limits of random constrained permutations. We show that for each $p\in (0,1)$, there are explicit constants $1/2 < α_*(p) \leq β^*(p) < 1$ such that the length of the longest increasing subsequence in a random permutation of size $n$ sampled from the Brownian separable permuton is between $n^{α_*(p) - o(1)}$ and $n^{β^*(p) + o(1)}$ with probability tending to 1 as $n\to\infty$. In the symmetric case $p=1/2$, we have $α_*(p) \approx 0.812$ and $β^*(p)\approx 0.975$. We present numerical simulations which suggest that the lower bound $α_*(p)$ is close to optimal in the whole range $p\in(0,1)$. Our results work equally well for the closely related Brownian cographons. In this setting, we show that for each $p\in (0,1)$, the size of the largest clique (resp. independent set) in a random graph on $n$ vertices sampled from the Brownian cographon is between $n^{α_*(p) - o(1)}$ and $n^{β^*(p) + o(1)}$ (resp. $n^{α_*(1-p) - o(1)}$ and $n^{β^*(1-p) + o(1)}$) with probability tending to 1 as $n\to\infty$. Our proofs are based on the analysis of a fragmentation process embedded in a Brownian excursion introduced by Bertoin (2002). We expect that our techniques can be extended to prove similar bounds for uniform separable permutations and uniform cographs.

math.PR↗

arXiv2023

Spatial growth-fragmentations and excursions from hyperplanes

In this paper, we are interested in the self-similar growth-fragmentation process that shows up when slicing half-space excursions of a $d$-dimensional Brownian motion from hyperplanes. Such a family of processes turns out to be a spatial self-similar growth-fragmentation processes driven by an isotropic self-similar Markov process. The former can be seen as multitype growth-fragmentation processes, in the sense of arXiv:2112.11091, where the set of types is $\mathbb{S}^{d-2}$, the $(d-1)$-dimensional unit sphere. In order to characterise such family of processes, we study their spinal description similarly as in the monotype and multitype settings. Finally, we extend our study to the case when the $d$-dimensional Brownian motion is replaced by an isotropic Markov process whose first $(d-1)$ coordinates are driven by an isotropic stable Lévy process and the remaining coordinate is an independent standard real-valued Brownian motion.

math.PR↗

arXiv2023

Multitype self-similar growth-fragmentation processes

In this paper, we are interested in multitype self-similar growth-fragmentation processes. More precisely, we investigate a multitype version of the self-similar growth-fragmentation processes introduced by Bertoin, therefore extending the signed case (considered by the first author in a previous work) to finitely many types. Our main result in this direction describes the law of the spine in the multitype setting. In order to do so, we introduce two genealogical martingales, in the same spirit as in the positive case, which allow us not only to obtain the law of the spine but also to study the limit of the empirical measure of fragments. We stress that our arguments only rely on the structure of the underlying Markov additive processes (MAPs), and hence is more general than the treatment of the signed case. Our methods also require new results on exponential functionals for MAPs and a multitype version of the tail estimates in multiplicative cascades which are interesting in their own right.

math.PR↗

arXiv2023

Self-similar signed growth-fragmentations

Growth-fragmentation processes model the evolution of positive masses which undergo binary divisions. The aim of this paper is twofold. First, we extend the theory of growth-fragmentation processes to allow signed mass. Among others, we introduce genealogical martingales and establish a spinal decomposition for the associated cell system, following arXiv:1605.00581. Then, we study a particular family of such self-similar signed growth-fragmentation processes which arise when cutting half-planar excursions at horizontal levels. When restricting this process to the positive masses, we recover part of the family introduced by Bertoin, Budd, Curien and Kortchemski in arXiv:1605.00581.

math.PR↗