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Using the randomized algorithm method developed by Duminil-Copin, Raoufi, Tassion (2019b) we exhibit sharp phase transition for the confetti percolation model. This provides an alternate proof that the critical parameter for percolation in this model is $1/2$ when the underlying shapes for the distinct colours arise from the same distribution and extends the work of Hirsch (2015) and M{u}ller (2016). In addition we study the covered area fraction for this model, which is akin to the covered volume fraction in continuum percolation. Modulo a certain `transitivity condition this study allows us to calculate exact critical parameter for percolation when the underlying shapes for different colours may be of different sizes. Similar results are also obtained for the Poisson Voronoi percolation model when different coloured points have different growth speeds.
We prove that the probability of crossing a large square in quenched Voronoi percolation converges to 1/2 at criticality, confirming a conjecture of Benjamini, Kalai and Schramm from 1999. The main new tools are a quenched version of the box-crossing
We study phase transition and percolation at criticality for three random graph models on the plane, viz., the homogeneous and inhomogeneous enhanced random connection models (RCM) and the Poisson stick model. These models are built on a homogeneous
In string percolation model, the study of colliding systems at high energies is based on a continuum percolation theory in two dimensions where the number of strings distributed in the surface of interest is strongly determined by the size and the en
We consider first-passage percolation with i.i.d. non-negative weights coming from some continuous distribution under a moment condition. We review recent results in the study of geodesics in first-passage percolation and study their implications for
The ellipses model is a continuum percolation process in which ellipses with random orientation and eccentricity are placed in the plane according to a Poisson point process. A parameter $alpha$ controls the tail distribution of the major axis distri