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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In this paper we study the convergence of dynamical discrete web (DyDW) to the dynamical Brownian web (DyBW) in the path space topology. We show that almost surely the DyBW has RCLL paths taking values in an appropriate metric space and as a sequence of RCLL paths, the scaled dynamical discrete web converges to the DyBW. This proves weak convergence of the DyDW process to the DyBW process

Department of Mathematics

World University root node

Convergence of the dynamical discrete web to the dynamical Brownian web

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).

Fulltext

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

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

Branching Brownian motion (BBM) is used as a population model where the location of a particle on the real line represents his or her fitness which evolves according to a Brownian motion due to mutations. In this context, the genealogical tree of the particles is important to understand the evolution of the population over time. The main goal of this article is to give an overview and motivations for some of the results on the genealogy of extremal particles of BBM. In the first part of this article, we discuss the results of L.-P. Arguin, A. Bovier and N. Kistler from [Genealogy of Extremal Particles of BBM, 2011] and [Poissonian Statistics in the Extremal Process of BBM, 2012]. Their articles give a partial answer to what the distribution of the limiting extremal process of BBM looks like. These two articles highlight, how genetical information can successfully be used to obtain rather extensive information for the limiting particle picture. In the second part, we discuss the results of J. Berestycki, N. Berestycki and J. Schweinsberg from [The Genealogy of BBM with Absorption, 2013]. BBM with absorption is related to a population model with selection. This article shows that the genealogy of the surviving particles (with a selective advantage), which are 'local' extremal particles, is then governed by a Bolthausen-Sznitman coalescent process. This result is in sharp contrast to the standard population models without selection.