Department of Mathematics

Ashoka University is a private university located in Sonipat, Haryana, India, focusing on liberal arts and sciences. It was established in 2014.&nbsp;It is guided by an academic council of academics and scholars from around the world. The council sets the university&#39;s academic standard, advising on matters of curriculum, faculty hiring and research.&nbsp;

Ashoka University offers a full-time 3 (+ 1) year undergraduate programme. The University comprises of&nbsp;2500 plus students, with a mix of both national and international students.&nbsp; &nbsp;

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Ashoka University

We extend expansion formulas of Liu given in 2013 to the context of multiple series over root systems. Liu and others have shown the usefulness of these formulas in Special Functions and number-theoretic contexts. We extend Wang and Ma's generalizations of Liu's work which they obtained using q-Lagrange inversion. We use the&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><msub is="true"><mrow is="true"><mi is="true">A</mi></mrow><mrow is="true"><mi is="true">n</mi></mrow></msub></math>&nbsp;and&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><msub is="true"><mrow is="true"><mi is="true">C</mi></mrow><mrow is="true"><mi is="true">n</mi></mrow></msub></math>&nbsp;Bailey transformation and other summation theorems due to Gustafson, Milne, Milne and Lilly, and others, from the theory of&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><msub is="true"><mrow is="true"><mi is="true">A</mi></mrow><mrow is="true"><mi is="true">n</mi></mrow></msub></math>,&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><msub is="true"><mrow is="true"><mi is="true">C</mi></mrow><mrow is="true"><mi is="true">n</mi></mrow></msub></math>&nbsp;and&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><msub is="true"><mrow is="true"><mi is="true">D</mi></mrow><mrow is="true"><mi is="true">n</mi></mrow></msub></math>&nbsp;basic <a href="https://www.sciencedirect.com/topics/mathematics/hypergeometric-series">hypergeometric series</a>.

World University root node

Advances in Applied Mathematics

Expansion formulas for multiple basic hypergeometric series over root systems

The two-dimensional directed spanning forest (DSF) introduced by Baccelli and Bordenave is a planar directed forest whose vertex set is given by a homogeneous Poisson point process N<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-caligraphic" mathvariant="script">N</mi></mrow></math>&nbsp;on R2<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">R</mi></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup></math>. If the DSF has direction −ey<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><msub><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mi>y</mi></mrow></msub></math>, the ancestor h(u)<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">u</mi></mrow></mrow><mo stretchy="false">)</mo></math>&nbsp;of a vertex u∈N<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">u</mi></mrow></mrow><mo>∈</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-caligraphic" mathvariant="script">N</mi></mrow></mrow></math>&nbsp;is the nearest Poisson point (in the L2<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msub></math>&nbsp;distance) having strictly larger y<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-coordinate. This construction induces complex geometrical dependencies. In this paper, we show that the collection of DSF paths, properly scaled, converges in distribution to the Brownian web (BW). This verifies a conjecture made by Baccelli and Bordenave in 2007 (Ann. Appl. Probab. 17 (2007) 305–359).The two-dimensional directed spanning forest (DSF) introduced by Baccelli and Bordenave is a planar directed forest whose vertex set is given by a homogeneous Poisson point process N<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-caligraphic" mathvariant="script">N</mi></mrow></math>&nbsp;on R2<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">R</mi></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup></math>. If the DSF has direction −ey<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><msub><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mi>y</mi></mrow></msub></math>, the ancestor h(u)<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">u</mi></mrow></mrow><mo stretchy="false">)</mo></math>&nbsp;of a vertex u∈N<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">u</mi></mrow></mrow><mo>∈</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-caligraphic" mathvariant="script">N</mi></mrow></mrow></math>&nbsp;is the nearest Poisson point (in the L2<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msub></math>&nbsp;distance) having strictly larger y<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-coordinate. This construction induces complex geometrical dependencies. In this paper, we show that the collection of DSF paths, properly scaled, converges in distribution to the Brownian web (BW). This verifies a conjecture made by Baccelli and Bordenave in 2007 (Ann. Appl. Probab. 17 (2007) 305–359).

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Annals of Probability

The 2d-directed spanning forest converges to the Brownian web

We study coexistence in discrete time multi-type frog models. We first show that for two types of particles on Zd, for d ≥ 2, for any jumping parameters p1, p2 ∈ (0, 1], coexistence occurs with positive probability for sufficiently rich deterministic initial configuration. We extend this to the case of random distribution of initial particles. We study the question of coexistence for multiple types and show positive probability coexistence of 2d types on Zd for rich enough initial configuration. We also show an instance of infinite coexistence on Zd for d ≥ 3 provided we have sufficiently rich initial configuration.

Electronic Communications in Probability

Coexistence in discrete time multi-type competing frog models

We prove decorrelation estimates for generalized lattice Anderson models on Zd constructed with finite-rank perturbations in the spirit of Klopp [12]. These are applied to prove that the local eigenvalue statistics ! ωE and ! ωE0 , associated with two energies E and E0 in the localization region and satisfying jE E0j > 4d, are independent. That is, if I; J are two bounded intervals, the random variables ! ωE.I / and ! ωE0.J /, are independent and distributed according to a compound Poisson distribution whose Levy measure has finite support. We also prove that the extended Minami estimate implies that the eigenvalues in the localization region have multiplicity at most the rank of the perturbation. The method of proof contains new ingredients that simplify the proof of the rank one case [12, 19, 21], extends tomodels forwhich the eigenvalues are degenerate, and applies tomodels forwhich the potential is not sign definite [20] in dimensions d > 1. © 2021 European Mathematical Society.

Journal of Spectral Theory

Decorrelation estimates for random Schrödinger operators with non rank one perturbations

We study the properties of stationary G-chains in terms of their generating functions. In particular, we prove an analogue of the Szego limit theorem for symplectic eigenvalues, derive an expression for the entropy rate of stationary quantum Gaussian processes, and study the distribution of symplectic eigenvalues of truncated block Toeplitz matrices. We also introduce a concept of symplectic numerical range, analogous to that of numerical range, and study some of its basic properties, mainly in the context of block Toeplitz operators. © 2021 European Mathematical Society Publishing House. All rights reserved.

A Szego type theorem and distribution of symplectic eigenvalues

If A is a real 2n×2n positive definite matrix, then there exists a symplectic matrix M such that MTAM=diag(D,D), where D is a positive diagonal matrix with diagonal entries d1(A)⩽⋯⩽dn(A). We prove a maxmin principle for dk(A) akin to the classical Courant–Fisher–Weyl principle for Hermitian eigenvalues and use it to derive an analogue of the Weyl inequality di+j−1(A+B)⩾di(A)+dj(B).

Canadian Mathematical Bulletin

Variational principles for symplectic eigenvalues

Stochastic networks based on random point sets as nodes have attracted considerable interest in many applications, particularly in communication networks, including wireless sensor networks, peer-to-peer networks and so on. The study of such networks generally requires the nodes to be independently and uniformly distributed as a Poisson point process. In this work, we venture beyond this standard paradigm and investigate the stochastic geometry of networks obtained from directed spanning forests (DSF) based on randomly perturbed lattices, which have desirable statistical properties as a models of spatially dependent point fields. In the regime of low disorder, we show in 2D and 3D that the DSF almost surely consists of a single tree. In 2D, we further establish that the DSF, as a collection of paths, converges under diffusive scaling to the Brownian web.

Journal of Statistical Physics

Transmission and Navigation on Disordered Lattice Networks, Directed Spanning Forests and Brownian Web

Convergence of directed forests, spanning on random subsets of lattices or on point processes, towards the Brownian web has made the subject of an abundant literature, a large part of which relies on a criterion proposed by Fontes, Isopi, Newman and Ravishankar (2004). One of their convergence condition, called (B2), states that the probability of the event that there exist three distinct paths for a time interval of length t (> 0), all starting within a segment of length ϵ, is of small order of ϵ. This condition is often verified by applying an FKG type correlation inequality together with a coalescing time tail estimate for two paths. For many models where paths have complex interactions, it is hard to establish FKG type inequalities. In this article, we show that for a non-crossing path model, with certain assumptions, a suitable upper bound on expected first collision time among three paths can be obtained directly using Lyapunov functions. This, in turn, provides an alternate verification of Condition (B2). We further show that in case of independent simple symmetric one-dimensional random walks or in case of independent Brownian motions, the expected value can be computed explicitly. We apply this alternate method of verification of (B2) to several models in the basin of attraction of the Brownian web studied earlier in the literature ([22], [14], [11]).

Lecture Notes Series, Institute for Mathematical Sciences

Collision Times of Random Walks and Applications to the Brownian Web

Let x and y be positive n-vectors. We show that there exists a 2n×2n positive definite real matrix whose symplectic spectrum is y, and the symplectic spectrum of whose diagonal is x if and only if x is weakly supermajorised by y. © 2020 Elsevier Inc.

Linear Algebra and Its Applications

A Schur-Horn theorem for symplectic eigenvalues

In this paper, we consider Anderson type operators on a separable Hilbert space where the random perturbations are finite rank and the random variables have full support on . We show that spectral multiplicity has a uniform lower bound whenever the lower bound is given on a set of positive Lebesgue measure on the point spectrum away from the continuous one. We also show a deep connection between the multiplicity of pure point spectrum and local spectral statistics, in particular, we show that spectral multiplicity higher than one always gives non-Poisson local statistics in the framework of Minami theory. In particular, for higher rank Anderson models with pure point spectrum, with the randomness having support equal to , there is a uniform lower bound on spectral multiplicity and in case this is larger than one, the local statistics is not Poisson. © 2020 World Scientific Publishing Company.