arXiv subjects

Elliot Paquette

Publications and source records attributed to Elliot Paquette.

65 records · Page 4Linked to original sources

arXiv2015

Extremal eigenvalue fluctuations in the GUE minor process and the law of fractional logarithm

We consider the GUE minor process, where a sequence of GUE matrices is drawn from the corner of a doubly infinite array of i.i.d. standard normal variables subject to the symmetry constraint. From each matrix, we take its largest eigenvalue, appropriately rescaled to converge to the standard Tracy-Widom distribution. We show the analogue of the law of iterated logarithm for this sequence, i.e. we divide the normalized n-th eigenvalue by a logarithmic factor and show the limsup of this sequence is a constant almost surely. We also give almost sure bounds for the appropriately scaled liminf.

math.PR↗

arXiv2015

Quantitative Small Subgraph Conditioning

We revisit the method of small subgraph conditioning, used to establish that random regular graphs are Hamiltonian a.a.s. We refine this method using new technical machinery for random $d$-regular graphs on $n$ vertices that hold not just asymptotically, but for any values of $d$ and $n$. This lets us estimate how quickly the probability of containing a Hamiltonian cycle converges to 1, and it produces quantitative contiguity results between different models of random regular graphs. These results hold with $d$ held fixed or growing to infinity with $n$. As additional applications, we establish the distributional convergence of the number of Hamiltonian cycles when $d$ grows slowly to infinity, and we prove that the number of Hamiltonian cycles can be approximately computed from the graph's eigenvalues for almost all regular graphs.

math.PR↗

arXiv2015

Birkhoff sum fluctuations in substitution dynamical systems

We consider the deviation of Birkhoff sums along fixed orbits of substitution dynamical systems. We show distributional convergence for the Birkhoff sums of eigenfunctions of the substitution matrix. For noncoboundary eigenfunctions with eigenvalue of modulus 1, we obtain a central limit theorem. For other eigenfunctions, we show convergence to distributions supported on Cantor sets. We also give a new criterion for such an eigenfunction to be a coboundary, as well as a new characterization of substitution dynamical systems with bounded discrepancy

math.DS↗

arXiv2014

Anchored expansion, speed, and the hyperbolic Poisson Voronoi tessellation

We show that random walk on a stationary random graph with positive anchored expansion and exponential volume growth has positive speed. We also show that two families of random triangulations of the hyperbolic plane, the hyperbolic Poisson Voronoi tessellation and the hyperbolic Poisson Delaunay triangulation, have 1-skeletons with positive anchored expansion. As a consequence, we show that the simple random walks on these graphs have positive speed. We include a section of open problems and conjectures on the topics of stationary geometric random graphs and the hyperbolic Poisson Voronoi tessellation.

math.PR↗

arXiv2014

Regularization of non-normal matrices by Gaussian noise

We consider the regularization of matrices $M^N$ written in Jordan form by additive Gaussian noise $N^{-γ}G^N$, where $G^N$ is a matrix of i.i.d. standard Gaussians and $γ>1/2$ so that the operator norm of the additive noise tends to $0$ with $N$. Under mild conditions on the structure of $M^N$ we evaluate the limit of the empirical measure of eigenvalues of $M^N+N^{-γ} G^N$ and show that it depends on $γ$, in contrast with the case of a single Jordan block.

math.PR↗

arXiv2014

The threshold for integer homology in random d-complexes

Let Y ~ Y_d(n,p) denote the Bernoulli random d-dimensional simplicial complex. We answer a question of Linial and Meshulam from 2003, showing that the threshold for vanishing of homology H_{d-1}(Y; Z) is less than 80d log n / n. This bound is tight, up to a constant factor.

math.AT↗

arXiv2014

The power of choice combined with preferential attachment

We prove almost sure convergence of the maximum degree in an evolving tree model combining local choice and preferential attachment. At each step in the growth of the graph, a new vertex is introduced. A fixed, finite number of possible neighbors are sampled from the existing vertices with probability proportional to degree. Of these possibilities, the vertex with the largest degree is chosen. The maximal degree in this model has linear or near-linear behavior. This contrasts sharply with what is seen in the same choice model without preferential attachment. The proof is based showing the tree has a persistent hub by comparison with the standard preferential attachment model, as well as martingale and random walk arguments.

math.PR↗

arXiv2014

The power of 2 choices over preferential attachment

We introduce a new type of preferential attachment tree that includes choices in its evolution, like with Achlioptas processes. At each step in the growth of the graph, a new vertex is introduced. Two possible neighbor vertices are selected independently and with probability proportional to degree. Between the two, the vertex with smaller degree is chosen, and a new edge is created. We determine with high probability the largest degree of this graph up to some additive error term.

math.PR↗

arXiv2012

Global Fluctuations for Linear Statistics of β-Jacobi Ensembles

We study the global fluctuations for linear statistics of the form $\sum_{i=1}^n f(λ_i)$ as $n \rightarrow \infty$, for $C^1$ functions $f$, and $λ_1, ..., λ_n$ being the eigenvalues of a (general) $β$-Jacobi ensemble, for which tridiagonal models were given by Killip and Nenciu as well as Edelman and Sutton. The fluctuation from the mean ($\sum_{i=1}^n f(λ_i) - \Exp \sum_{i=1}^n f(λ_i)$) is given asymptotically by a Gaussian process. We compute the covariance matrix for the process and show that it is diagonalized by a shifted Chebyshev polynomial basis; in addition, we analyze the deviation from the predicted mean for polynomial test functions, and we obtain a law of large numbers.

math.PR↗

arXiv2012

Functional limit theorems for random regular graphs

Consider d uniformly random permutation matrices on n labels. Consider the sum of these matrices along with their transposes. The total can be interpreted as the adjacency matrix of a random regular graph of degree 2d on n vertices. We consider limit theorems for various combinatorial and analytical properties of this graph (or the matrix) as n grows to infinity, either when d is kept fixed or grows slowly with n. In a suitable weak convergence framework, we prove that the (finite but growing in length) sequences of the number of short cycles and of cyclically non-backtracking walks converge to distributional limits. We estimate the total variation distance from the limit using Stein's method. As an application of these results we derive limits of linear functionals of the eigenvalues of the adjacency matrix. A key step in this latter derivation is an extension of the Kahn-Szemerédi argument for estimating the second largest eigenvalue for all values of d and n.

math.PR↗

arXiv2008

The index of a vector field on an orbifold with boundary

A Poincaré-Hopf theorem in the spirit of Pugh is proven for compact orbifolds with boundary. The theorem relates the index sum of a smooth vector field in generic contact with the boundary orbifold to the Euler-Satake characteristic of the orbifold and a boundary term. The boundary term is expressed as a sum of Euler characteristics of tangency and exit-region orbifolds. As a corollary, we express the index sum of the vector field induced on the inertia orbifold to the Euler characteristics of the associated underlying topological spaces.

math.DG↗