arXiv subjects

Mike Liu

Publications and source records attributed to Mike Liu.

6 recordsLinked to original sources

arXiv2026

Uniqueness and locality of the ground state of the disordered Monomer-Dimer models on independently weighted Unimodular Bienaym\'e-Galton-Watson trees

Consider a finite graph $G=(V(G),E(G))$ and two continuous weight distributions $\omega$ and $\xi$, for which we only assume that $\xi$ is lower bounded. Next, independently draw weights $(w(e))_{e \in E(G)}$ with distribution $\omega$ on edges and $(x(v))_{v \in V(G)}$ with distribution $\xi$ on vertices. The ground state of the monomer-dimer model on the weighted graph $G$ is a collection of edges (dimers) and vertices (monomers) such that every vertex is included in at most one monomer or dimer, and such that the sum of weights on its dimers and monomers is maximised. Take $(G_n,o_n)_{n \in \mathbb{N}}$ to be a sequence of random rooted weighted graphs that converges locally to an independently weighted unimodular Bienaym\'e-Galton-Watson tree $(\mathbb{T},o)$ with vertex-weight distribution $\xi$ and edge-weight distribution $\omega$ . By proving that the ground state of the monomer-dimer model on the tree $(\mathbb{T},o)$ is almost surely unique and locally approximable, we prove that the ground state of the monomer-dimer model on $(G_n,o_n)$ must converge locally to the ground state of the monomer-dimer model on $(\mathbb{T},o)$. This also implies a strong decorrelation property on monomer-dimer models on unimodular Bienaym\'e-Galton-Watson trees.

math.PR↗

arXiv2026

Optimal matching under size priority

Past studies on the local limit of maximal weight matchings in edge-weighted large random graphs rely fundamentally on the assumption that the weights are atomless, which ensures that the maximal weight matching is unique. This excludes de facto maximal size matchings that correspond to equal edge-weights. In this work, we overcome this difficulty by assigning i.i.d.~atomless weights to edges and choosing the maximal size matching that maximises the weight. We call these doubly constrained matchings \emph{optimal matchings}. The natural generalisation of optimal matchings for infinite unimodular random graphs are unimodular matchings of maximal density at the root that maximise the expected weight at the root when it is matched. For unimodular Bienaym\'e-Galton-Watson (UBGW) trees and for a broad class of weight distributions, we show existence and uniqueness in law of such matchings. We also prove that if a sequence of finite random weighted graphs converges locally to an UBGW tree with i.i.d.~weights, then there exists a sequence of matchings on the finite graphs that converges locally to the optimal matching on the limiting tree. Finally, we identify a regime, depending only on the offspring distribution of the limiting tree, in which correlations between edge states in the optimal matching decay exponentially with their graph distance. In this regime, we strengthen the previous convergence to the convergence of optimal matchings of the finite graphs. As a by-product, we can explicitly compute the asymptotic densities of edges that belong to all maximal-density matchings, and of edges that belong to none.

math.PR↗

arXiv2025

Percolative properties of the random coprime colouring

Given $u$ and $v$ in $\mathbb{Z}^d$, say that $u$ is visible from $v$ if the segment from $u$ to $v$ contains exactly two elements, which are $u$ and $v$. Take $X$ "uniformly at random in $\mathbb{Z}^d$" and colour each vertex $u$ of $\mathbb{Z}^d$ in white if $u$ is visible from $X$ and in black otherwise. Previous independent works of Pleasants-Huck and of the third author give a precise meaning to this definition. This paper is dedicated to the study of this random colouring from the point of view of percolation theory: given a reasonable graph structure on $\mathbb{Z}^d$, how many infinite black (resp. white) connected components are there?

math.PR↗

arXiv2024

Optimal Unimodular Matching

We consider sequences of finite weighted random graphs that converge locally to unimodular i.i.d. weighted random trees. When the weights are atomless, we prove that the matchings of maximal weight converge locally to a matching on the limiting tree. For this purpose, we introduce and study unimodular matchings on weighted unimodular random trees as well as a notion of optimality for these objects. In this context, we prove that, in law, there is a unique optimal unimodular matching for a given unimodular tree. We then prove that this law is the local limit of the sequence of matchings of maximal weight. Along the way, we also show that this law is characterised by an equation derived from a message passing algorithm.

math.PR↗

arXiv2023

Simulating medium-spectral-resolution exoplanet characterization with SCALES angular/reference differential imaging

SCALES (Slicer Combined with Array of Lenslets for Exoplanet Spectroscopy) is a 2 - 5 micron high-contrast lenslet-based integral field spectrograph (IFS) designed to characterize exoplanets and their atmospheres. The SCALES medium-spectral-resolution mode uses a lenslet subarray with a 0.34 x 0.36 arcsecond field of view which allows for exoplanet characterization at increased spectral resolution. We explore the sensitivity limitations of this mode by simulating planet detections in the presence of realistic noise sources. We use the SCALES simulator scalessim to generate high-fidelity mock observations of planets that include speckle noise from their host stars, as well as other atmospheric and instrumental noise effects. We employ both angular and reference differential imaging as methods of disentangling speckle noise from the injected planet signals. These simulations allow us to assess the feasibility of speckle deconvolution for SCALES medium resolution data, and to test whether one approach outperforms another based on planet angular separations and contrasts.

astro-ph.IM↗

arXiv2008

Performance of the Near-infrared coronagraphic imager on Gemini-South

We present the coronagraphic and adaptive optics performance of the Gemini-South Near-Infrared Coronagraphic Imager (NICI). NICI includes a dual-channel imager for simultaneous spectral difference imaging, a dedicated 85-element curvature adaptive optics system, and a built-in Lyot coronagraph. It is specifically designed to survey for and image large extra-solar gaseous planets on the Gemini Observatory 8-meter telescope in Chile. We present the on-sky performance of the individual subsystems along with the end-to-end contrast curve. These are compared to our model predictions for the adaptive optics system, the coronagraph, and the spectral difference imaging.

astro-ph↗