arXiv subjects

Antti Knowles

Publications and source records attributed to Antti Knowles.

At least 19 recordsLinked to original sources

arXiv2026

Semilocalization for inhomogeneous random graphs

We analyse the eigenvectors of the adjacency matrix of a random inhomogeneous graph constructed from a specified degree sequence. We assume that the empirical degree sequence has bounded mean and variance. We show that near the edges of the spectrum, the eigenvectors are semilocalized in the sense that their mass concentrates around a small set of resonant vertices. For the extremal eigenvalues, we establish localization around a single vertex. In order to obtain effective estimates in the presence of highly inhomogeneous degrees, we introduce a new economical pruning procedure that carefully extracts a forest from the original graph, whose adjacency matrix is compared to that of the original graph using a suitably constructed local coupling to random trees with independent edges.

math.PR↗

arXiv2026

The Euclidean $\phi^4_2$ theory as a limit of an inhomogeneous Bose gas

We prove that the grand canonical Gibbs state of an interacting two-dimensional quantum Bose gas confined by a trapping potential converges to the complex Euclidean field theory with local quartic self-interaction, when the density of the gas becomes large and the range of the interaction becomes small. We obtain convergence of the relative partition function and convergence in $L^1 \cap L^\infty$ of the renormalised reduced density matrices. The field theory is supported on distributions of negative regularity, which requires a renormalisation by divergent mass and energy counterterms. Unlike previous results in the homogeneous setting of the torus without a trapping potential, the counterterms are not given by a finite collection of scalars but by diverging counterterm functions. This leads to significant new mathematical challenges. For our proof, we also derive quantitative bounds on the Green function of Schr\"odinger operators and of its gradient, which might be of independent interest.

math-ph↗

arXiv2023

Localized phase for the Erd\H{o}s-R\'enyi graph

We analyse the eigenvectors of the adjacency matrix of the Erd\H{o}s-R\'enyi graph $\mathbb G(N,d/N)$ for $\sqrt{\log N} \ll d \lesssim \log N$. We show the existence of a localized phase, where each eigenvector is exponentially localized around a single vertex of the graph. This complements the completely delocalized phase previously established in [arXiv:2005.14180]. For large enough $d$, we establish a mobility edge by showing that the localized phase extends up to the boundary of the delocalized phase. We derive explicit asymptotics for the localization length up to the mobility edge and characterize its divergence near the phase boundary. The proof is based on a rigorous verification of Mott's criterion for localization, comparing the tunnelling amplitude between localization centres with the eigenvalue spacing. The first main ingredient is a new family of global approximate eigenvectors, for which sharp enough estimates on the tunnelling amplitude can be established. The second main ingredient is a lower bound on the spacing of approximate eigenvalues. It follows from an anticoncentration result for these discrete random variables, obtained by a recursive application of a self-improving anticoncentration estimate due to Kesten.

math.PR↗

arXiv2023

The Euclidean $ϕ^4_2$ theory as a limit of an interacting Bose gas

We prove that the complex Euclidean field theory with local quartic self-interaction in two dimensions arises as a limit of an interacting Bose gas at positive temperature, when the density of the gas becomes large and the range of the interaction becomes small. The field theory is supported on distributions of negative regularity, which requires a renormalization by divergent mass and energy counterterms. We obtain convergence of the relative partition function and uniform convergence of the renormalized reduced density matrices. The proof is based on three main ingredients: (a) a quantitative analysis of the infinite-dimensional saddle point argument for the functional integral introduced in [32] using continuity properties of Brownian paths, (b) a Nelson-type estimate for a general nonlocal field theory in two dimensions, and (c) repeated Gaussian integration by parts in field space to obtain uniform control on the renormalized correlation functions. As a byproduct of our proof, in two and three dimensions we also extend the results on the mean-field limit from [32,56] to unbounded interaction potentials satisfying the optimal integrability conditions proposed by Bourgain [13].

math-ph↗

arXiv2022

Poisson statistics and localization at the spectral edge of sparse Erdős-Rényi graphs

We consider the adjacency matrix $A$ of the Erdős-Rényi graph on $N$ vertices with edge probability $d/N$. For $(\log \log N)^4 \ll d \lesssim \log N$, we prove that the eigenvalues near the spectral edge form asymptotically a Poisson process and the associated eigenvectors are exponentially localized. As a corollary, at the critical scale $d \asymp \log N$, the limiting distribution of the largest nontrivial eigenvalue does not match with any previously known distribution. Together with [arXiv:2005.14180], our result establishes the coexistence of a fully delocalized phase and a fully localized phase in the spectrum of $A$. The proof relies on a three-scale rigidity argument, which characterizes the fluctuations of the eigenvalues in terms of the fluctuations of sizes of spheres of radius 1 and 2 around vertices of large degree.

math.PR↗

arXiv2021

The completely delocalized region of the Erdős-Rényi graph

We analyse the eigenvectors of the adjacency matrix of the Erdős-Rényi graph on $N$ vertices with edge probability $\frac{d}{N}$. We determine the full region of delocalization by determining the critical values of $\frac{d}{\log N}$ down to which delocalization persists: for $\frac{d}{\log N} > \frac{1}{\log 4 - 1}$ all eigenvectors are completely delocalized, and for $\frac{d}{\log N} > 1$ all eigenvectors with eigenvalues away from the spectral edges are completely delocalized. Below these critical values, it is known [arXiv:2005.14180, arXiv:2106.12519] that localized eigenvectors exist in the corresponding spectral regions.

math.PR↗

arXiv2021

Delocalization transition for critical Erdős-Rényi graphs

We analyse the eigenvectors of the adjacency matrix of a critical Erdős-Rényi graph $\mathbb G(N,d/N)$, where $d$ is of order $\log N$. We show that its spectrum splits into two phases: a delocalized phase in the middle of the spectrum, where the eigenvectors are completely delocalized, and a semilocalized phase near the edges of the spectrum, where the eigenvectors are essentially localized on a small number of vertices. In the semilocalized phase the mass of an eigenvector is concentrated in a small number of disjoint balls centred around resonant vertices, in each of which it is a radial exponentially decaying function. The transition between the phases is sharp and is manifested in a discontinuity in the localization exponent $γ(\mathbf w)$ of an eigenvector $\mathbf w$, defined through $\|\mathbf w\|_\infty / \|\mathbf w\|_2 = N^{-γ(\mathbf w)}$. Our results remain valid throughout the optimal regime $\sqrt{\log N} \ll d \leq O(\log N)$.

math.PR↗

arXiv2021

The mean-field limit of quantum Bose gases at positive temperature

We prove that the grand canonical Gibbs state of an interacting quantum Bose gas converges to the Gibbs measure of a nonlinear Schrödinger equation in the mean-field limit, where the density of the gas becomes large and the interaction strength is proportional to the inverse density. Our results hold in dimensions $d \leq 3$. For $d > 1$ the Gibbs measure is supported on distributions of negative regularity and we have to renormalize the interaction. More precisely, we prove the convergence of the relative partition function and of the reduced density matrices in the $L^r$-norm with optimal exponent $r$. Moreover, we prove the convergence in the $L^\infty$-norm of Wick-ordered reduced density matrices, which allows us to control correlations of Wick-ordered particle densities as well as the asymptotic distribution of the particle number. Our proof is based on a functional integral representation of the grand canonical Gibbs state, in which convergence to the mean-field limit follows formally from an infinite-dimensional stationary phase argument for ill-defined non-Gaussian measures. We make this argument rigorous by introducing a white-noise-type auxiliary field, through which the functional integral is expressed in terms of propagators of heat equations driven by time-dependent periodic random potentials and can, in turn, be expressed as a gas of interacting Brownian loops and paths. When the gas is confined by an external trapping potential, we control the decay of the reduced density matrices using excursion probabilities of Brownian bridges.

math-ph↗

arXiv2021

Fluctuations of extreme eigenvalues of sparse Erdős-Rényi graphs

We consider a class of sparse random matrices which includes the adjacency matrix of the Erdős-Rényi graph $\mathcal{G}(N,p)$. We show that if $N^{\varepsilon} \leq Np \leq N^{1/3-\varepsilon}$ then all nontrivial eigenvalues away from 0 have asymptotically Gaussian fluctuations. These fluctuations are governed by a single random variable, which has the interpretation of the total degree of the graph. This extends the result [19] on the fluctuations of the extreme eigenvalues from $Np \geq N^{2/9 + \varepsilon}$ down to the optimal scale $Np \geq N^{\varepsilon}$. The main technical achievement of our proof is a rigidity bound of accuracy $N^{-1/2-\varepsilon} \, (Np)^{-1/2}$ for the extreme eigenvalues, which avoids the $(Np)^{-1}$-expansions from [9,19,24]. Our result is the last missing piece, added to [8, 12, 19, 24], of a complete description of the eigenvalue fluctuations of sparse random matrices for $Np \geq N^{\varepsilon}$.

math.PR↗

arXiv2021

Spectral radii of sparse random matrices

We establish bounds on the spectral radii for a large class of sparse random matrices, which includes the adjacency matrices of inhomogeneous Erdős-Rényi graphs. Our error bounds are sharp for a large class of sparse random matrices. In particular, for the Erdős-Rényi graph $G(n,d/n)$, our results imply that the smallest and second-largest eigenvalues of the adjacency matrix converge to the edges of the support of the asymptotic eigenvalue distribution provided that $d \gg \log n$. Together with the companion paper [3], where we analyse the extreme eigenvalues in the complementary regime $d \ll \log n$, this establishes a crossover in the behaviour of the extreme eigenvalues around $d \sim \log n$. Our results also apply to non-Hermitian sparse random matrices, corresponding to adjacency matrices of directed graphs. The proof combines (i) a new inequality between the spectral radius of a matrix and the spectral radius of its nonbacktracking version together with (ii) a new application of the method of moments for nonbacktracking matrices.

math.PR↗

arXiv2020

Interacting loop ensembles and Bose gases

We study interacting Bose gases in thermal equilibrium on a lattice. We establish convergence of the grand canonical Gibbs states of such gases to their mean-field (classical field) and large-mass (classical particle) limits. The former is a classical field theory for a complex scalar field with quartic self-interaction. The latter is a classical theory of point particles with two-body interactions. Our analysis is based on representations in terms of ensembles of interacting random loops, the Ginibre loop ensemble for Bose gases and the Symanzik loop ensemble for classical scalar field theories. For small enough interactions, our results also hold in infinite volume.

math-ph↗

arXiv2020

Edge rigidity and universality of random regular graphs of intermediate degree

For random $d$-regular graphs on $N$ vertices with $1 \ll d \ll N^{2/3}$, we develop a $d^{-1/2}$ expansion of the local eigenvalue distribution about the Kesten-McKay law up to order $d^{-3}$. This result is valid up to the edge of the spectrum. It implies that the eigenvalues of such random regular graphs are more rigid than those of Erdős-Rényi graphs of the same average degree. As a first application, for $1 \ll d \ll N^{2/3}$, we show that all nontrivial eigenvalues of the adjacency matrix are with very high probability bounded in absolute value by $(2 + o(1)) \sqrt{d - 1}$. As a second application, for $N^{2/9} \ll d \ll N^{1/3}$, we prove that the extremal eigenvalues are concentrated at scale $N^{-2/3}$ and their fluctuations are governed by Tracy-Widom statistics. Thus, in the same regime of $d$, $52\%$ of all $d$-regular graphs have second-largest eigenvalue strictly less than $2 \sqrt{d - 1}$.

math.PR↗

arXiv2020

Extremal eigenvalues of critical Erdős-Rényi graphs

We complete the analysis of the extremal eigenvalues of the the adjacency matrix $A$ of the Erdős-Rényi graph $G(N,d/N)$ in the critical regime $d \asymp \log N$ of the transition uncovered in [arXiv:1704.02953,arXiv:1704.02945], where the regimes $d \gg \log N$ and $d \ll \log N$ were studied. We establish a one-to-one correspondence between vertices of degree at least $2d$ and nontrivial (excluding the trivial top eigenvalue) eigenvalues of $A / \sqrt{d}$ outside of the asymptotic bulk $[-2,2]$. This correspondence implies that the transition characterized by the appearance of the eigenvalues outside of the asymptotic bulk takes place at the critical value $d = d_* = \frac{1}{\log 4 - 1} \log N$. For $d < d_*$ we obtain rigidity bounds on the locations of all eigenvalues outside the interval $[-2,2]$, and for $d > d_*$ we show that no such eigenvalues exist. All of our estimates are quantitative with polynomial error probabilities. Our proof is based on a tridiagonal representation of the adjacency matrix and on a detailed analysis of the geometry of the neighbourhood of the large degree vertices. An important ingredient in our estimates is a matrix inequality obtained via the associated nonbacktracking matrix and an Ihara-Bass formula [arXiv:1704.02945]. Our argument also applies to sparse Wigner matrices, defined as the Hadamard product of $A$ and a Wigner matrix, in which case the role of the degrees is replaced by the squares of the $\ell^2$-norms of the rows.

math.PR↗

arXiv2020

A path-integral analysis of interacting Bose gases and loop gases

We review some recent results on interacting Bose gases in thermal equilibrium. In particular, we study the convergence of the grand-canonical equilibrium states of such gases to their mean-field limits, which are given by the Gibbs measures of classical field theories with quartic Hartree-type self-interaction, and to the Gibbs states of classical gases of point particles. We discuss various open problems and conjectures concerning, e.g., Bose-Einstein condensation, polymers and $\vert \boldsymbolϕ \vert^{4}$-theory.

math-ph↗

arXiv2019

Mesoscopic eigenvalue density correlations of Wigner matrices

We investigate to what extent the microscopic Wigner-Gaudin-Mehta-Dyson (WGMD) (or sine kernel) statistics of random matrix theory remain valid on mesoscopic scales. To that end, we compute the connected two-point spectral correlation function of a Wigner matrix at two mesoscopically separated points. In the mesoscopic regime, density correlations are much weaker than in the microscopic regime. Our result is an explicit formula for the two-point function. This formula implies that the WGMD statistics are valid to leading order on all mesoscopic scales, that in the real symmetric case there are subleading corrections matching precisely the WGMD statistics, while in the complex Hermitian case these subleading corrections are absent. We also uncover non-universal subleading correlations, which dominate over the universal ones beyond a certain intermediate mesoscopic scale. The proof is based on a hierarchy of Schwinger-Dyson equations for a sufficiently large class of polynomials in the entries of the Green function. The hierarchy is indexed by a tree, whose depth is controlled using stopping rules. A key ingredient in the derivation of the stopping rules is a new estimate on the density of states, which we prove to have bounded derivatives of all order on all mesoscopic scales.

math.PR↗

arXiv2019

Gibbs measures of nonlinear Schrödinger equations as limits of many-body quantum states in dimensions $d \leq 3$

We prove that Gibbs measures of nonlinear Schrödinger equations arise as high-temperature limits of thermal states in many-body quantum mechanics. Our results hold for defocusing interactions in dimensions $d =1,2,3$. The many-body quantum thermal states that we consider are the grand canonical ensemble for $d = 1$ and an appropriate modification of the grand canonical ensemble for $d =2,3$. In dimensions $d =2,3$, the Gibbs measures are supported on singular distributions, and a renormalization of the chemical potential is necessary. On the many-body quantum side, the need for renormalization is manifested by a rapid growth of the number of particles. We relate the original many-body quantum problem to a renormalized version obtained by solving a counterterm problem. Our proof is based on ideas from field theory, using a perturbative expansion in the interaction, organized by using a diagrammatic representation, and on Borel resummation of the resulting series.

math-ph↗

arXiv2019

Local law and complete eigenvector delocalization for supercritical Erdős-Rényi graphs

We prove a local law for the adjacency matrix of the Erdős-Rényi graph $G(N, p)$ in the supercritical regime $ pN \geq C\log N$ where $G(N,p)$ has with high probability no isolated vertices. In the same regime, we also prove the complete delocalization of the eigenvectors. Both results are false in the complementary subcritical regime. Our result improves the corresponding results from [11] by extending them all the way down to the critical scale $pN = O(\log N)$. A key ingredient of our proof is a new family of multilinear large deviation estimates for sparse random vectors, which carefully balance mixed $\ell^2$ and $\ell^\infty$ norms of the coefficients with combinatorial factors, allowing us to prove strong enough concentration down to the critical scale $pN = O(\log N)$. These estimates are of independent interest and we expect them to be more generally useful in the analysis of very sparse random matrices.

math.PR↗

arXiv2018

Lectures on the local semicircle law for Wigner matrices

These notes provide an introduction to the local semicircle law from random matrix theory, as well as some of its applications. We focus on Wigner matrices, Hermitian random matrices with independent upper-triangular entries with zero expectation and constant variance. We state and prove the local semicircle law, which says that the eigenvalue distribution of a Wigner matrix is close to Wigner's semicircle distribution, down to spectral scales containing slightly more than one eigenvalue. This local semicircle law is formulated using the Green function, whose individual entries are controlled by large deviation bounds. We then discuss three applications of the local semicircle law: first, complete delocalization of the eigenvectors, stating that with high probability the eigenvectors are approximately flat; second, rigidity of the eigenvalues, giving large deviation bounds on the locations of the individual eigenvalues; third, a comparison argument for the local eigenvalue statistics in the bulk spectrum, showing that the local eigenvalue statistics of two Wigner matrices coincide provided the first four moments of their entries coincide. We also sketch further applications to eigenvalues near the spectral edge, and to the distribution of eigenvectors.

math.PR↗