arXiv subjects

Adrien Kassel

Publications and source records attributed to Adrien Kassel.

15 recordsLinked to original sources

arXiv2026

Trace identities for quiver representations

We give an expression for the determinant of the twisted Laplacian associated with any linear representation of a finite quiver in terms of traces of the holonomy of its cycles. To establish this expression, we prove a general identity for the determinant of a block matrix in terms of traces of products of its blocks. We give two proofs, one purely enumerative and one using generating series. In the special case of a finite graph equipped with a vector bundle and a connection, the twisted Laplacian determinant admits a combinatorial interpretation as a weighted count of tuples of oriented cycle-rooted spanning forests, where the weights involve traces of holonomies along cycles formed by combining the edges of the forests.

math.CO↗

arXiv2023

On the mean projection theorem for determinantal point processes

In this short note, we extend to the continuous case a mean projection theorem for discrete determinantal point processes associated with a finite range projection, thus strengthening a known result in random linear algebra due to Ermakov and Zolotukhin. We also give a new formula for the variance of the exterior power of the random projection.

math.PR↗

arXiv2023

Graph coverings and twisted operators

Given a graph and a representation of its fundamental group, there is a naturally associated twisted adjacency operator. The main result of this article is the fact that these operators behave in a controlled way under graph covering maps. When such an operator can be used to enumerate objects, or compute a partition function, this has concrete implications on the corresponding enumeration problem, or statistical mechanics model. For example, we show that if $\widetildeΓ$ is a finite connected covering graph of a graph $Γ$ endowed with edge-weights $x=\{x_e\}_e$, then the spanning tree partition function of $Γ$ divides the one of $\widetildeΓ$ in the ring $\mathbb{Z}[x]$. Several other consequences are obtained, some known, others new.

math.CO↗

arXiv2022

Determinantal random subgraphs

We define two families of determinantal random spanning subgraphs of a finite connected graph, one supported by acyclic spanning subgraphs (spanning forests) with fixed number of connected components, the other by connected spanning subgraphs with fixed number of independent cycles. Each family generalizes the uniform spanning tree and the generating functions of these probability measures generalize the classical Kirchhoff and Symanzik polynomials. We call Symanzik spanning forests the elements of the acyclic spanning subgraphs family, and single out a particular determinantal mixture of these, having as kernel a normalized Laplacian on $1$-forms, which we call the Laplacian spanning forest. Our proofs rely on a set of integral and real or complex (which we call geometric) multilinear identies involving cycles, coboundaries, and forests on graphs. We prove these identities using classical pieces of the algebraic topology of graphs and the exterior calculus applied to finite determinantal point processes, both of which we treat in a self-contained way. We emphasize the matroidal nature of our constructions, thereby showing how the above two families of random spanning subgraphs are dual to one another, as well as possible generalisations.

math.PR↗

arXiv2020

Determinantal probability measures on Grassmannians

We introduce and study a class of determinantal probability measures generalising the class of discrete determinantal point processes. These measures live on the Grassmannian of a real, complex, or quaternionic inner product space that is split into pairwise orthogonal finite-dimensional subspaces. They are determined by a positive self-adjoint contraction of the inner product space, in a way that is equivariant under the action of the group of isometries that preserve the splitting.

math.PR↗

arXiv2020

Covariant Symanzik identities

Classical isomorphism theorems due to Dynkin, Eisenbaum, Le Jan, and Sznitman establish equalities between the correlation functions or distributions of occupation times of random paths or ensembles of paths and Markovian fields, such as the discrete Gaussian free field. We extend these results to the case of real, complex, or quaternionic vector bundles of arbitrary rank over graphs endowed with a connection, by providing distributional identities between functionals of the Gaussian free vector field and holonomies of random paths. As an application, we give a formula for computing moments of a large class of random, in general non-Gaussian, fields in terms of holonomies of random paths with respect to an annealed random gauge field, in the spirit of Symanzik's foundational work on the subject.

math.PR↗

arXiv2019

A colourful path to matrix-tree theorems

In this short note, we revisit Zeilberger's proof of the classical matrix-tree theorem and give a unified concise proof of variants of this theorem, some known and some new.

math.CO↗

arXiv2017

Active spanning trees with bending energy on planar maps and SLE-decorated Liouville quantum gravity for $κ> 8$

We introduce a two-parameter family of probability measures on spanning trees of a planar map. One of the parameters controls the activity of the spanning tree and the other is a measure of its bending energy. When the bending parameter is 1, we recover the active spanning tree model, which is closely related to the critical Fortuin--Kasteleyn model. A random planar map decorated by a spanning tree sampled from our model can be encoded by means of a generalized version of Sheffield's hamburger-cheeseburger bijection. Using this encoding, we prove that for a range of parameter values (including the ones corresponding to maps decorated by an active spanning tree), the infinite-volume limit of spanning-tree-decorated planar maps sampled from our model converges in the peanosphere sense, upon rescaling, to an SLE$_κ$-decorated $γ$-Liouville quantum cone with $κ> 8$ and $γ= 4/\sqrtκ\in (0,\sqrt 2)$.

math.PR↗

arXiv2017

Revisiting the combinatorics of the 2D Ising model

We provide a concise exposition with original proofs of combinatorial formulas for the 2D Ising model partition function, multi-point fermionic observables, spin and energy density correlations, for general graphs and interaction constants, using the language of Kac-Ward matrices. We also give a brief account of the relations between various alternative formalisms which have been used in the combinatorial study of the planar Ising model: dimers and Grassmann variables, spin and disorder operators, and, more recently, s-holomorphic observables. In addition, we point out that these formulas can be extended to the double-Ising model, defined as a pointwise product of two Ising spin configurations on the same discrete domain, coupled along the boundary.

math.CO↗

arXiv2017

Random curves on surfaces induced from the Laplacian determinant

We define natural probability measures on cycle-rooted spanning forests (CRSFs) on graphs embedded on a surface with a Riemannian metric. These measures arise from the Laplacian determinant and depend on the choice of a unitary connection on the tangent bundle to the surface. We show that, for a sequence of graphs $(G_n)$ conformally approximating the surface, the measures on CRSFs of $G_n$ converge and give a limiting probability measure on finite multicurves (finite collections of pairwise disjoint simple closed curves) on the surface, independent of the approximating sequence. Wilson's algorithm for generating spanning trees on a graph generalizes to a cycle-popping algorithm for generating CRSFs for a general family of weights on the cycles. We use this to sample the above measures. The sampling algorithm, which relates these measures to the loop-erased random walk, is also used to prove tightness of the sequence of measures, a key step in the proof of their convergence. We set the framework for the study of these probability measures and their scaling limits and state some of their properties.

math.PR↗

arXiv2016

Active spanning trees and Schramm-Loewner evolution

We consider the Peano curve separating a spanning tree from its dual spanning tree on an embedded planar graph, where the tree and dual tree are weighted by $y$ to the number of active edges, and "active" is in the sense of the Tutte polynomial. When the graph is a portion of the square grid approximating a simply connected domain, it is known ($y=1$ and $y=1+\sqrt{2}$) or believed ($1<y<3$) that the Peano curve converges to a space-filling SLE$_κ$ loop, where $y=1-2\cos(4π/κ)$, corresponding to $4<κ\leq 8$. We argue that the same should hold for $0\le y<1$, which corresponds to $8<κ\leq 12$.

cond-mat.stat-mech↗

arXiv2015

Transfer current and pattern fields in spanning trees

When a simply connected domain $D\subset{\mathbb{R}}^d$ ($d\ge 2$) is approximated in a "good" way by embedded connected weighted graphs, we prove that the transfer current matrix (defined on the edges of the graph viewed as an electrical network) converges, up to a local weight factor, to the differential of Green's function on $D$. This observation implies that properly rescaled correlations of the spanning tree model and correlations of minimal subconfigurations in the abelian sandpile model have a universal and conformally covariant limit. We further show that, on a periodic approximation of the domain, all pattern fields of the spanning tree model, as well as the minimal-pattern (e.g. zero-height) fields of the sandpile, converge weakly in distribution to Gaussian white noise.

math.PR↗

arXiv2015

The looping rate and sandpile density of planar graphs

We give a simple formula for the looping rate of loop-erased random walk on a finite planar graph. The looping rate is closely related to the expected amount of sand in a recurrent sandpile on the graph. The looping rate formula is well-suited to taking limits where the graph tends to an infinite lattice, and we use it to give an elementary derivation of the (previously computed) looping rate and sandpile densities of the square, triangular, and honeycomb lattices, and compute (for the first time) the looping rate and sandpile densities of many other lattices, such as the kagome lattice, the dice lattice, and the truncated hexagonal lattice (for which the values are all rational), and the square-octagon lattice (for which it is transcendental).

math.PR↗

arXiv2014

Random two-component spanning forests

We study random two-component spanning forests ($2$SFs) of finite graphs, giving formulas for the first and second moments of the sizes of the components, vertex-inclusion probabilities for one or two vertices, and the probability that an edge separates the components. We compute the limit of these quantities when the graph tends to an infinite periodic graph in ${\mathbb R}^d$.

math.PR↗

arXiv2014

The singularity probability of random diagonally-dominant Hermitian matrices

In this note we describe the singular locus of diagonally-dominant Hermitian matrices with nonnegative diagonal entries over the reals, the complex numbers, and the quaternions. This yields explicit expressions for the probability that such matrices, chosen at random, are singular. For instance, in the case of the identity $n \times n$ matrix perturbed by a symmetric, zero-diagonal, $\{\pm 1/(n-1)\}$-Bernoulli matrix, this probability turns out to be equal to $2^{-(n-1)(n-2)/2}$. As a corollary, we find that the probability for a $n\times n$ symmetric $\{\pm 1\}$-Bernoulli matrix to have eigenvalue $n$ is $2^{-(n^2-n+2)/2}$.

math.PR↗