arXiv subjects

Anirban Basak

Publications and source records attributed to Anirban Basak.

At least 19 recordsLinked to original sources

arXiv2025

Spectral properties of random perturbations of non-normal Toeplitz matrices

Spectral properties of Toeplitz operators and their finite truncations have long been central in operator theory. In the finite dimensional, non-normal setting, the spectrum is notoriously unstable under perturbations. Random perturbations provide a natural framework for studying this instability and identifying spectral features that emerge in typical noisy situations. This article surveys recent advances on the spectral behavior of (polynomially vanishing) random perturbations of Toeplitz matrices, focusing mostly on the limiting spectral distribution, the distribution of outliers, and localization of eigenvectors, and highlight the major techniques used to study these problems. We complement the survey with new results on the limiting spectral distribution of polynomially vanishing random perturbation of Toeplitz matrices with continuous symbols, on the limiting spectral distribution of finitely banded Toeplitz matrices under exponentially and super-exponentially vanishing random perturbations, and on the complete localization of outlier eigenvectors of randomly perturbed Jordan blocks.

math.PR↗

arXiv2025

Upper tail bounds for irregular graphs

We consider the upper tail large deviations of subgraph counts for irregular graphs $\mathrm{H}$ in $\mathbb{G}(n,p)$, the sparse Erd\H{o}s-R\'enyi graph on $n$ vertices with edge connectivity probability $p \in (0,1)$. For $n^{-1/\Delta} \ll p \ll 1$, where $\Delta$ is the maximum degree of $\mathrm{H}$, we derive the upper tail large deviations for any irregular graph $\mathrm{H}$. On the other hand, we show that for $p$ such that $1 \ll n^{v_{\mathrm{H}}} p^{e_{\mathrm{H}}} \ll (\log n)^{\alpha^{*}_{\mathrm{H}}/\left(\alpha^{*}_{\mathrm{H}}-1\right)}$, where $v_{\mathrm{H}}$ and $e_{\mathrm{H}}$ denote the number of vertices and edges of $\mathrm{H}$, and $\alpha^*_{\mathrm{H}}$ denotes the fractional independence number, the upper tail large deviations of the number of unlabelled copies of $\mathrm{H}$ in $\mathbb{G}(n,p)$ is given by that of a sequence of Poisson random variables with diverging mean, for any strictly balanced graph $\mathrm{H}$. Restricting to the $r$-armed star graph we further prove a localized behavior in the intermediate range of $p$ (left open by the above two results) and show that the mean-field approximation is asymptotically tight for the logarithm of the upper tail probability. This work further identifies the typical structures of $\mathbb{G}(n,p)$ conditioned on upper tail rare events in the localized regime.

math.PR↗

arXiv2023

Potts and random cluster measures on locally regular-tree-like graphs

Fixing $\beta \ge 0$ and an integer $q \ge 2$, consider the ferromagnetic $q$-Potts measures $\mu_n^{\beta,B}$ on finite graphs ${\sf G}_n$ on $n$ vertices, with external field strength $B \ge 0$ and the corresponding random cluster measures $\varphi^{q,\beta,B}_{n}$. Suppose that as $n \to \infty$ the uniformly sparse graphs ${\sf G}_n$ converge locally to an infinite $d$-regular tree ${\sf T}_{d}$, $d \ge 3$. We show that the convergence of the Potts free energy density to its Bethe replica symmetric prediction (which has been proved in case $d$ is even, or when $B=0$), yields the local weak convergence of $\varphi^{q,\beta,B}_n$ and $\mu_n^{\beta,B}$ to the corresponding free or wired random cluster measure, Potts measure, respectively, on ${\sf T}_{d}$. The choice of free versus wired limit is according to which has the larger Potts Bethe functional value, with mixtures of these two appearing {as limit points on} the critical line $\beta_c(q,B)$ where these two values of the Bethe functional coincide. For $B=0$ and $\beta>\beta_c$, we further establish a pure-state decomposition by showing that conditionally on the same dominant color $1 \le k \le q$, the $q$-Potts measures on such edge-expander graphs ${\sf G}_n$ converge locally to the $q$-Potts measure on ${\sf T}_{d}$ with a boundary wired at color $k$.

math.PR↗

arXiv2023

Upper tail of the spectral radius of sparse Erdős-Rényi graphs

We consider an Erdős-Rényi graph $\mathbb{G}(n,p)$ on $n$ vertices with edge probability $p$ such that \[ \sqrt{\frac{\log n}{\log \log n}} \ll np \le n^{1/2-o(1)}, \label{eq:abs} \tag{$\dagger$} \] and derive the upper tail large deviations of $λ(\mathbb{G}(n,p))$, the largest eigenvalue of its adjacency matrix. Within this regime we show that, for $p \gg n^{-2/3}$ the $\log$-probability of the upper tail event of $λ(\mathbb{G}(n,p))$ equals to that of planting a clique of an appropriate size (upon ignoring smaller order terms), while for $p \ll n^{-2/3}$ the same is given by that of the existence of a high degree vertex. This, in particular, shows an emergence of {\em non-planted localized structure} in the latter regime. We also confirm that in the entire regime \eqref{eq:abs} the large deviation probability is asymptotically approximated by the solution of the mean-field variational problem, and further identify the typical structure of $\mathbb{G}(n,p)$ conditioned on the upper tail event of $λ(\mathbb{G}(n,p))$ in a certain sub-regime of $p$. For $p$ such that $\log(np) \gtrsim \log n$ the large deviations of $λ(\mathbb{G}(n,p))$ is deduced from those of ${\rm Hom}(C_{2t}, \mathbb{G}(n,p))$, the homomorphism counts of cycle of length $2t$, for $t \ge 3$ and $p$ such that $n^{1/2-o(1)} \ge np \gg n^{1/t}$. In this latter regime the typical structure of $\mathbb{G}(n,p)$ conditioned on the upper tail of ${\rm Hom}(C_{2t}, \mathbb{G}(n,p))$ is identified and asymptotic tightness of the mean-field approximation is also established.

math.PR↗

arXiv2023

Large deviations of the largest eigenvalue of supercritical sparse Wigner matrices

Consider a random symmetric matrix with i.i.d.~entries on and above its diagonal that are products of Bernoulli random variables and random variables with sub-Gaussian tails. Such a matrix will be called a sparse Wigner matrix and can be viewed as the adjacency matrix of a random network with sub-Gaussian weights on its edges. In the regime where the mean degree is at least logarithmic in dimension, the edge eigenvalues of an appropriately scaled sparse Wigner matrix stick to the edges of the support of the semicircle law. We show that in this sparsity regime, the large deviations upper tail event of the largest eigenvalue of a sparse Wigner matrix with sub-Gaussian entries is generated by either the emergence of a high degree vertex with a large vertex weight or that of a clique with large edge weights. Interestingly, the rate function obtained is discontinuous at the typical value of the largest eigenvalue, which accounts for the fact that its large deviation behaviour is generated by finite rank perturbations. This complements the results of Ganguly and Nam, and Ganguly, Hiesmayr, and Nam which considered the case where the mean degree is constant.

math.PR↗

arXiv2022

Robust Subthermionic Topological Transistor Action via Antiferromagnetic Exchange

The topological quantum field-effect transition in buckled 2D-Xenes can potentially be engineered to enable sub-thermionic transistor operation coupled with dissipationless ON-state conduction. Substantive device design strategies to harness this will necessitate delving into the physics of the quantum field effect transition between the dissipationless topological phase and the band insulator phase. Investigating workable device structures, we uncover fundamental sub-threshold limits posed by the gating mechanism that effectuates such a transition, thereby emphasizing the need for innovations on materials and device structures. Detailing the complex band translation physics related to the quantum spin Hall effect phase transition, it is shown that a gating strategy to beat the thermionic limit can be engineered at the cost of sacrificing the dissipationless ON-state conduction. It is then demonstrated that an out-of-plane antiferromagnetic exchange introduced in the material via proximity coupling can incite transitions between the quantum spin-valley Hall and the spin quantum anomalous Hall phase, which can ultimately ensure the topological robustness of the ON state while surpassing the thermionic limit. Our work thus underlines the operational criteria for building topological transistors using quantum materials that can overcome the Boltzmann's tyranny while preserving the topological robustness.

cond-mat.mes-hall↗

arXiv2021

Momentum relaxation effects in 2D-Xene field effect device structures

We analyze the electric field driven topological field effect transition on 2D-xene materials with the addition of momentum relaxation effects, in order to account for dephasing processes. The topological field effect transition between the quantum spin Hall phase and the quantum valley Hall phase is analyzed in detail using the Keldysh non-equilibrium Green's function technique with the inclusion of momentum and phase relaxation, within the self-consistent Born approximation. Details of the transition with applied electric field are elucidated for the ON-OFF characteristics with emphasis on the transport properties along with the tomography of the current carrying edge states. We note that for moderate momentum relaxation, the current carrying quantum spin Hall edge states are still pristine and show moderate decay with propagation. To facilitate our analysis, we introduce two metrics in our calculations, the coherent transmission and the effective transmission. In elucidating the physics clearly, we show that the effective transmission, which is derived rigorously from the quantum mechanical current operator is indeed the right quantity to analyze topological stability against dephasing. Exploring further, we show that the insulating quantum valley Hall phase, as a result of dephasing carries band-tails which potentially activates parasitic OFF currents, thereby degrading the ON-OFF ratios. Our analysis sets the stage for realistic modeling of topological field effect devices for various applications, with the inclusion of scattering effects and analyzing their role in the optimization of the device performance.

cond-mat.mes-hall↗

arXiv2021

Localization of eigenvectors of non-Hermitian banded noisy Toeplitz matrices

We prove localization with high probability on sets of size of order $N/\log N$ for the eigenvectors of non-Hermitian finitely banded $N\times N$ Toeplitz matrices $P_N$ subject to small random perturbations, in a very general setting. As perturbation we consider $N\times N$ random matrices with independent entries of zero mean, finite moments, and which satisfy an appropriate anti-concentration bound. We show via a Grushin problem that an eigenvector for a given eigenvalue $z$ is well approximated by a random linear combination of the singular vectors of $P_N-z$ corresponding to its small singular values. We prove precise probabilistic bounds on the local distribution of the eigenvalues of the perturbed matrix and provide a detailed analysis of the singular vectors to conclude the localization result.

math.SP↗

arXiv2021

Sharp transition of the invertibility of the adjacency matrices of sparse random graphs

We consider three different models of sparse random graphs:~undirected and directed Erdős-Rényi graphs, and random bipartite graph with an equal number of left and right vertices. For such graphs we show that if the edge connectivity probability $p \in (0,1)$ satisfies $n p \ge \log n + k(n)$ with $k(n) \to \infty$ as $n \to \infty$, then the adjacency matrix is invertible with probability approaching one (here $n$ is the number of vertices in the two former cases and the number of left and right vertices in the latter case). If $np \le \log n -k(n)$ then these matrices are invertible with probability approaching zero, as $n \to \infty$. In the intermediate region, when $np=\log n + k(n)$, for a bounded sequence $k(n) \in \mathbb{R}$, the event $Ω_0$ that the adjacency matrix has a zero row or a column and its complement both have non-vanishing probability. For such choices of $p$ our results show that conditioned on the event $Ω_0^c$ the matrices are again invertible with probability tending to one. This shows that the primary reason for the non-invertibility of such matrices is the existence of a zero row or a column. The bounds on the probability of the invertibility of these matrices are a consequence of quantitative lower bounds on their smallest singular values. Combining this with an upper bound on the largest singular value of the centered version of these matrices we show that the (modified) condition number is $O(n^{1+o(1)})$ on the event that there is no zero row or column, with large probability. This matches with von Neumann's prediction about the condition number of random matrices up to a factor of $n^{o(1)}$, for the entire range of $p$.

math.PR↗

arXiv2020

Outliers of random perturbations of Toeplitz matrices with finite symbols

Consider an $N\times N$ Toeplitz matrix $T_N$ with symbol ${a }(λ) := \sum_{\ell=-d_2}^{d_1} a_\ell λ^\ell$, perturbed by an additive noise matrix $N^{-γ} E_N$, where the entries of $E_N$ are centered i.i.d.~random variables of unit variance and $γ>1/2$. It is known that the empirical measure of eigenvalues of the perturbed matrix converges weakly, as $N\to\infty$, to the law of ${a}(U)$, where $U$ is distributed uniformly on $\mathbb{S}^1$. In this paper, we consider the outliers, i.e. eigenvalues that are at a positive ($N$-independent) distance from ${a}(\mathbb{S}^1)$. We prove that there are no outliers outside ${\rm spec} \, T({a})$, the spectrum of the limiting Toeplitz operator, with probability approaching one, as $N \to \infty$. {In contrast,} in ${\rm spec}\, T({a})\setminus {a}({\mathbb S}^1)$ the process of outliers converges to the point process described by the zero set of certain random {analytic} functions. The limiting random {analytic} functions can be expressed as linear combinations of the determinants of finite sub-matrices of an infinite dimensional matrix, whose entries are i.i.d.~having the same law as that of $E_N$. The coefficients in the linear combination depend on the roots of the polynomial $P_{z, {a}}(λ):= ({a}(λ) -z)λ^{d_2}=0$ and semi-standard Young Tableaux with shapes determined by the number of roots of $P_{z,{a}}(λ)=0$ that are greater than one in moduli.

math.PR↗

arXiv2020

Upper tail large deviations of regular subgraph counts in Erdős-Rényi graphs in the full localized regime

For a $Δ$-regular connected graph ${\sf H}$ the problem of determining the upper tail large deviation for the number of copies of ${\sf H}$ in $\mathbb{G}(n,p)$, an Erdős-Rényi graph on $n$ vertices with edge probability $p$, has generated significant interests. For $p\ll 1$ and $np^{Δ/2} \gg (\log n)^{1/(v_{\sf H}-2)}$, where $v_{\sf H}$ is the number of vertices in ${\sf H}$, the upper tail large deviation event is believed to occur due to the presence of localized structures. In this regime the large deviation event that the number of copies of ${\sf H}$ in $\mathbb{G}(n,p)$ exceeds its expectation by a constant factor is predicted to hold at a speed $n^2 p^Δ \log (1/p)$ and the rate function is conjectured to be given by the solution of a mean-field variational problem. After a series of developments in recent years, covering progressively broader ranges of $p$, the upper tail large deviations for cliques of fixed size was proved by Harel, Mousset, and Samotij \cite{hms} in the entire localized regime. This paper establishes the conjecture for all connected regular graphs in the whole localized regime.

math.PR↗

arXiv2019

Spectrum of random perturbations of Toeplitz matrices with finite symbols

Let $T_N$ denote an $N\times N$ Toeplitz matrix with finite, $N$ independent symbol ${\bf a}$. For $E_N$ a noise matrix satisfying mild assumptions (ensuring, in particular, that $N^{-1/2}\|E_N\|_{\rm HS}\to_{N\to\infty} 0$ at a polynomial rate), we prove that the empirical measure of eigenvalues of $T_N+E_N$ converges to the law of ${\bf a}(U)$, where $U$ is uniformly distributed on the unit circle in the complex plane. This extends results from arXiv:1712.00042 to the non-triangular setup and non complex Gaussian noise, and confirms predictions obtained in Reichel and Trefethen (1992) using the notion of pseudo-spectrum.

math.PR↗

arXiv2018

Regularization of non-normal matrices by Gaussian noise - the banded Toeplitz and twisted Toeplitz cases

We consider the spectrum of additive, polynomially vanishing random perturbations of deterministic matrices, as follows. Let $M_N$ be a deterministic $N\times N$ matrix, and let $G_N$ be a complex Ginibre matrix. We consider the matrix $\mathcal{M}_N=M_N+N^{-γ}G_N$, where $γ>1/2$. With $L_N$ the empirical measure of eigenvalues of $\mathcal{M}_N$, we provide a general deterministic equivalence theorem that ties $L_N$ to the singular values of $z-M_N$, with $z\in \mathbb{C}$. We then compute the limit of $L_N$ when $M_N$ is an upper triangular Toeplitz matrix of finite symbol: if $M_N=\sum_{i=0}^{\mathfrak{d}} a_i J^i$ where $\mathfrak{d}$ is fixed, $a_i\in\mathcal{C}$ are deterministic scalars and $J$ is the nilpotent matrix $J(i,j)={\bf 1}_{j=i+1}$, then $L_N$ converges, as $N\to\infty$, to the law of $\sum_{i=0}^{\mathfrak{d}} a_i U^i$ where $U$ is a uniform random variable on the unit circle in the complex plane. We also consider the case of slowly varying diagonals (twisted Toeplitz matrices), and, when $\mathfrak{d}=1$, also of i.i.d.~entries on the diagonals in $M_N$.

math.PR↗

arXiv2018

The circular law for sparse non-Hermitian matrices

For a class of sparse random matrices of the form $A_n =(ξ_{i,j}δ_{i,j})_{i,j=1}^n$, where $\{ξ_{i,j}\}$ are i.i.d.~centered sub-Gaussian random variables of unit variance, and $\{δ_{i,j}\}$ are i.i.d.~Bernoulli random variables taking value $1$ with probability $p_n$, we prove that the empirical spectral distribution of $A_n/\sqrt{np_n}$ converges weakly to the circular law, in probability, for all $p_n$ such that $p_n=ω({\log^2n}/{n})$. Additionally if $p_n$ satisfies the inequality $np_n > \exp(c\sqrt{\log n})$ for some constant $c$, then the above convergence is shown to hold almost surely. The key to this is a new bound on the smallest singular value of complex shifts of real valued sparse random matrices. The circular law limit also extends to the adjacency matrix of a directed Erdős-Rényi graph with edge connectivity probability $p_n$.

math.PR↗

arXiv2018

Diffusion limit for the partner model at the critical value

The partner model is an SIS epidemic in a population with random formation and dissolution of partnerships, and with disease transmission only occuring within partnerships. Foxall, Edwards, and van den Driessche found the critical value and studied the subcritical and supercritical regimes. Recently Foxall has shown that (if there are enough initial infecteds $I_0$) the extinction time in the critical model is of order $\sqrt{N}$. Here we improve that result by proving the convergence of $i_N(t)=I(\sqrt{N}t)/\sqrt{N}$ to a limiting diffusion. We do this by showing that within a short time, this four dimensional process collapses to two dimensions: the number of $SI$ and $II$ partnerships are constant multiples of the the number of infected singles. The other variable, the total number of singles, fluctuates around its equilibrium like an Ornstein-Uhlenbeck process of magnitude $\sqrt{N}$ on the original time scale and averages out of the limit theorem for $i_N(t)$. As a by-product of our proof we show that if $τ_N$ is the extinction time of $i_N(t)$ (on the $\sqrt{N}$ time scale) then $τ_N$ has a limit.

math.PR↗

arXiv2018

Circular law for the sum of random permutation matrices

Let $P_n^1,\dots, P_n^d$ be $n\times n$ permutation matrices drawn independently and uniformly at random, and set $S_n^d:=\sum_{\ell=1}^d P_n^\ell$. We show that if $\log^{12}n/(\log \log n)^{4} \le d=O(n)$, then the empirical spectral distribution of $S_n^d/\sqrt{d}$ converges weakly to the circular law in probability as $n \to \infty$.

math.PR↗

arXiv2017

Invertibility of Sparse non-Hermitian matrices

We consider a class of sparse random matrices of the form $A_n =(ξ_{i,j}δ_{i,j})_{i,j=1}^n$, where $\{ξ_{i,j}\}$ are i.i.d.~centered random variables, and $\{δ_{i,j}\}$ are i.i.d.~Bernoulli random variables taking value $1$ with probability $p_n$, and prove a quantitative estimate on the smallest singular value for $p_n = Ω(\frac{\log n}{n})$, under a suitable assumption on the spectral norm of the matrices. This establishes the invertibility of a large class of sparse matrices. For $p_n =Ω( n^{-α})$ with some $α\in (0,1)$, we deduce that the condition number of $A_n$ is of order $n$ with probability tending to one under the optimal moment assumption on $\{ξ_{i,j}\}$. This in particular, extends a conjecture of von Neumann about the condition number to sparse random matrices with heavy-tailed entries. In the case that the random variables $\{ξ_{i,j}\}$ are i.i.d.~sub-Gaussian, we further show that a sparse random matrix is singular with probability at most $\exp(-c n p_n)$ whenever $p_n$ is above the critical threshold $p_n = Ω(\frac{ \log n}{n})$. The results also extend to the case when $\{ξ_{i,j}\}$ have a non-zero mean. We further find quantitative estimates on the smallest singular value of the adjacency matrix of a directed Erdős-Réyni graph whenever its edge connectivity probability is above the critical threshold $Ω(\frac{\log n}{n})$.

math.PR↗

arXiv2016

Large subgraphs in pseudo-random graphs

We consider classes of pseudo-random graphs on $n$ vertices for which the degree of every vertex and the co-degree between every pair of vertices are in the intervals $(np - Cn^δ,np+Cn^δ)$ and $(np^2- C n^δ, np^2 +C n^δ)$ respectively, for some absolute constant $C$, and $p, δ\in (0,1)$. We show that for such pseudo-random graphs the number of induced isomorphic copies of subgraphs of size $s$ are approximately same as that of an Erdős-Réyni random graph with edge connectivity probability $p$ as long as $s \le (((1-δ)\wedge \frac{1}{2})-o(1))\log n/\log (1/p)$, when $p \in (0,1/2]$. When $p \in (1/2,1)$ we obtain a similar result. Our result is applicable for a large class of random and deterministic graphs including exponential random graph models (ERGMs), thresholded graphs from high-dimensional correlation networks, Erdős-Réyni random graphs conditioned on large cliques, random $d$-regular graphs and graphs obtained from vector spaces over binary fields. In the context of the last example, the results obtained are optimal. Straight-forward extensions using the proof techniques in this paper imply strengthening of the above results in the context of larger motifs if a model allows control over higher co-degree type functionals.

math.PR↗