arXiv subjects

Riddhipratim Basu

Publications and source records attributed to Riddhipratim Basu.

46 records · Page 3Linked to original sources

arXiv2016

Last Passage Percolation with a Defect Line and the Solution of the Slow Bond Problem

We address the question of how a localized microscopic defect, especially if it is small with respect to certain dynamic parameters, affects the macroscopic behavior of a system. In particular we consider two classical exactly solvable models: Ulam's problem of the maximal increasing sequence and the totally asymmetric simple exclusion process. For the first model, using its representation as a Poissonian version of directed last passage percolation on $\mathbb R^2$, we introduce the defect by placing a positive density of extra points along the diagonal line. For the latter, the defect is produced by decreasing the jump rate of each particle when it crosses the origin. The powerful algebraic tools for studying these processes break down in the perturbed versions of the models. Taking a more geometric approach we show that in both cases the presence of an arbitrarily small defect affects the macroscopic behavior of the system: in Ulam's problem the time constant increases, and for the exclusion process the flux of particles decreases. This, in particular, settles the longstanding Slow Bond Problem.

math.PR↗

arXiv2015

Games on Random Boards

We consider the following two-player game on a graph. A token is located at a vertex, and the players take turns to move it along an edge to a vertex that has not been visited before. A player who cannot move loses. We analyze outcomes with optimal play on percolation clusters of Euclidean lattices. On Z^2 with two different percolation parameters for odd and even sites, we prove that the game has no draws provided closed sites of one parity are sufficiently rare compared with those of the other parity (thus favoring one player). We prove this also for certain d-dimensional lattices with d>=3. It is an open question whether draws can occur when the two parameters are equal. On a finite ball of Z^2, with only odd sites closed but with the external boundary consisting of even sites, we identify up to logarithmic factors a critical window for the trade-off between the size of the ball and the percolation parameter. Outside this window, one or other player has a decisive advantage. Our analysis of the game is intimately tied to the effect of boundary conditions on maximum-cardinality matchings.

math.PR↗

arXiv2015

Characterization of cutoff for reversible Markov chains

A sequence of Markov chains is said to exhibit (total variation) cutoff if the convergence to stationarity in total variation distance is abrupt. We consider reversible lazy chains. We prove a necessary and sufficient condition for the occurrence of the cutoff phenomena in terms of concentration of hitting time of "worst" (in some sense) sets of stationary measure at least $α$, for some $α\in (0,1)$. We also give general bounds on the total variation distance of a reversible chain at time $t$ in terms of the probability that some "worst" set of stationary measure at least $α$ was not hit by time $t$. As an application of our techniques we show that a sequence of lazy Markov chains on finite trees exhibits a cutoff iff the ratio of their relaxation-times and their (lazy) mixing-times tends to 0.

math.PR↗

arXiv2015

Evolving Voter Model on Dense Random Graphs

In this paper we examine a variant of the voter model on a dynamically changing network where agents have the option of changing their friends rather than changing their opinions. We analyse, in the context of dense random graphs, two models considered in Durrett et. al.(Proc. Natl. Acad. Sci. 109: 3682-3687, 2012). When an edge with two agents holding different opinion is updated, with probability $\fracβ{n}$, one agent performs a voter model step and changes its opinion to copy the other, and with probability $1-\fracβ{n}$, the edge between them is broken and reconnected to a new agent chosen randomly from (i) the whole network (rewire-to-random model) or, (ii) the agents having the same opinion (rewire-to-same model). We rigorously establish in both the models, the time for this dynamics to terminate exhibits a phase transition in the model parameter $β$. For $β$ sufficiently small, with high probability the network rapidly splits into two disconnected communities with opposing opinions, whereas for $β$ large enough the dynamics runs for longer and the density of opinion changes significantly before the process stops. In the rewire-to-random model, we show that a positive fraction of both opinions survive with high probability.

math.PR↗

arXiv2014

Bi-Lipschitz Expansion of Measurable Sets

We show that for $0<γ, γ' <1$ and for measurable subsets of the unit square with Lebesgue measure $γ$ there exist bi-Lipschitz maps with bounded Lipschitz constant (uniformly over all such sets) which are identity on the boundary and increases the Lebesgue measure of the set to at least $1-γ'$.

math.AP↗

arXiv2014

Scheduling of non-colliding random walks

On the complete graph ${\cal{K}}_M$ with $M \ge3$ vertices consider two independent discrete time random walks $\mathbb{X}$ and $\mathbb{Y}$, choosing their steps uniformly at random. A pair of trajectories $\mathbb{X} = \{ X_1, X_2, \dots \}$ and $\mathbb{Y} = \{Y_1, Y_2, \dots \}$ is called {\it{non-colliding}}, if by delaying their jump times one can keep both walks at distinct vertices forever. It was conjectured by P. Winkler that for large enough $M$ the set of pairs of non-colliding trajectories $\{\mathbb{X},\mathbb{Y} \} $ has positive measure. N. Alon translated this problem to the language of coordinate percolation, a class of dependent percolation models, which in most situations is not tractable by methods of Bernoulli percolation. In this representation Winkler's conjecture is equivalent to the existence of an infinite open cluster for large enough $M$. In this paper we establish the conjecture.

math.PR↗

arXiv2012

Lipschitz embeddings of random sequences

We develop a new multi-scale framework flexible enough to solve a number of problems involving embedding random sequences into random sequences. Grimmett, Liggett and Richthammer asked whether there exists an increasing M-Lipschitz embedding from one i.i.d. Bernoulli sequences into an independent copy with positive probability. We give a positive answer for large enough M. A closely related problem is to show that two independent Poisson processes on R are roughly isometric (or quasi-isometric). Our approach also applies in this case answering a conjecture of Szegedy and of Peled. Our theorem also gives a new proof to Winkler's compatible sequences problem.

math.PR↗

arXiv2012

Joint convergence of several copies of different patterned random matrices

We study the joint convergence of independent copies of several patterned matrices in the noncommutative probability setup. In particular, joint convergence holds for the well known Wigner, Toeplitz, Hankel, reverse circulant and symmetric circulant matrices. We also study some properties of the limits. In particular, we show that copies of Wigner becomes asymptotically free with copies of any of the above other matrices.

math.PR↗

arXiv2012

Spectral properties of random triangular matrices

We prove the existence of the limiting spectral distribution (LSD) of symmetric triangular patterned matrices and also establish the joint convergence of sequences of such matrices. For the particular case of the symmetric triangular Wigner matrix, we derive expression for the moments of the LSD using properties of Catalan words. The problem of deriving explicit formulae for the moments of the LSD does not seem to be easy to solve for other patterned matrices. The LSD of the non-symmetric triangular Wigner matrix also does not seem to be easy to establish.

math.PR↗

arXiv2011

Limiting Spectral Distribution of Block Matrices with Toeplitz Block Structure

We study two specific symmetric random block Toeplitz (of dimension $k \times k$) matrices: where the blocks (of size $n \times n$) are (i) matrices with i.i.d. entries, and (ii) asymmetric Toeplitz matrices. Under suitable assumptions on the entries, their limiting spectral distributions (LSDs) exist (after scaling by $\sqrt{nk}$) when (a) $k$ is fixed and $n \to\infty$ (b) $n$ is fixed and $k\rightarrow \infty$ (c) $n$ and $k$ go to $\infty$ simultaneously. Further the LSD's obtained in (a) and (b) coincide with those in (c) when $n$ or respectively $k$ tends to infinity. This limit in (c) is the semicircle law in case (i). In Case (ii) the limit is related to the limit of the random symmetric Toepiltz matrix as obtained by Bryc et al.(2006) and Hammond and Miller(2005).

math.PR↗