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Maxim Krikun

Publications and source records attributed to Maxim Krikun.

26 records · Page 2Linked to original sources
arXiv2008

Phase transition for the Ising model on the Critical Lorentzian triangulation

Ising model without external field on an infinite Lorentzian triangulation sampled from the uniform distribution is considered. We prove uniqueness of the Gibbs measure in the high temperature region and coexistence of at least two Gibbs measures at low temperature. The proofs are based on the disagreement percolation method and on a variant of Peierls method. The critical temperature is shown to be constant a.s.

math.PR↗
arXiv2007

Stochastic Models for Phylogenetic Trees on Higher-order Taxa

Simple stochastic models for phylogenetic trees on species have been well studied. But much paleontology data concerns time series or trees on higher-order taxa, and any broad picture of relationships between extant groups requires use of higher-order taxa. A coherent model for trees on (say) genera should involve both a species-level model and a model for the classification scheme by which species are assigned to genera. We present a general framework for such models, and describe three alternate classification schemes. Combining with the species-level model of Aldous-Popovic (2005), one gets models for higher-order trees, and we initiate analytic study of such models. In particular we derive formulas for the lifetime of genera, for the distribution of number of species per genus, and for the offspring structure of the tree on genera.

q-bio.PE↗
arXiv2007

Connected allocation to Poisson points in R^2

This note answers one question in [math.PR/0505668], concerning the connected allocation for the Poisson process in R^2. The proposed solution makes use of the Riemann map from the plane minus the minimal spanning forest of the Poisson point process to the halfplane. A picture of a numerically simulated example is included.

math.PR↗
arXiv2006

Local structure of random quadrangulations

This paper is an adaptation of a method used in \cite{K} to the model of random quadrangulations. We prove local weak convergence of uniform measures on quadrangulations and show that the local growth of quadrangulation is governed by certain critical time-reversed branching process and the rescaled profile converges to the reversed continuous-state branching process. As an intermediate result we derieve a biparametric generating function for certain class of quadrangulations with boundary.

math.PR↗
arXiv2005

Percolating paths through random points :

We prove consistency of four different approaches to formalizing the idea of minimum average edge-length in a path linking some infinite subset of points of a Poisson process. The approaches are (i) shortest path from origin through some $m$ distinct points; (ii) shortest average edge-length in paths across the diagonal of a large cube; (iii) shortest path through some specified proportion $δ$ of points in a large cube; (iv) translation-invariant measures on paths in $\Reals^d$ which contain a proportion $δ$ of the Poisson points. We develop basic properties of a normalized average length function $c(δ)$ and pose challenging open problem

math.PR↗
arXiv2003

Uniform infinite planar triangulation and related time-reversed critical branching process

We establish a connection between the uniform infinite planar triangulation and some critical time-reversed branching process. This allows to find a scaling limit for the principal boundary component of a ball of radius R for large R (i.e. for a boundary component separating the ball from infinity). We show also that outside of R-ball a contour exists that has length linear in R.

math.PR↗