Related papers: Stochastic Dynamics of Growing Young Diagrams and …
We systematically study the boundaries of one-dimensional, 2-color cellular automata depending on 4 cells, begun from simple initial conditions. We determine the exact growth rates of the boundaries that appear to be reducible. Morphic…
In this work we are considering the behavior of the limit shape of Young diagrams associated to random permutations on the set $\{1,\dots,n\}$ under a particular class of multiplicative measures. Our method is based on generating functions…
It is well known that the different cell-division models, such as Timer, Sizer, and Adder, can be distinguished based on the correlations between different single-cell-level quantities such as birth-size, division-time, division-size, and…
The dynamics of linear stochastic growth equations on growing substrates is studied. The substrate is assumed to grow in time following the power law $t^\gamma$, where the growth index $\gamma$ is an arbitrary positive number. Two different…
Growth-fragmentation processes model systems of cells that grow continuously over time and then fragment into smaller pieces. Typically, on average, the number of cells in the system exhibits asynchronous exponential growth and, upon…
The branching rule for the tower of wreath products of a finite group by the symmetric groups induces a stochastic process on the set of multiple Young diagrams through random transitions of boxes of the diagrams between one another. We…
Cells achieve size homeostasis by regulating their division timing based on their size, added size, and cell cycle time. Previous research under steady-state conditions demonstrated the robustness of these mechanisms. However, their dynamic…
We investigate the large deviations of the shape of the random RSK Young diagrams associated with a random word of size $n$ whose letters are independently drawn from an alphabet of size $m=m(n)$. When the letters are drawn uniformly and…
The complexity of gene regulatory networks in multicellular organisms makes interpretable low-dimensional models highly desirable. An attractive geometric picture, attributed to Waddington, visualizes the differentiation of a cell into…
Locally finite self-similar graphs with bounded geometry and without bounded geometry as well as non-locally finite self-similar graphs are characterized by the structure of their cell graphs. Geometric properties concerning the volume…
In this paper we compute the precise asymptotics of the variance of linear statistic of descents on a growing interval for Plancherel Young diagrams (following Vershik and Kerov, diagrams are considered rotated by $\pi/4$). We also give an…
Establishing a quantitative connection between the population growth rate and the generation times of single cells is a prerequisite for understanding evolutionary dynamics of microbes. However, existing theories fail to account for the…
Given a random word of size $n$ whose letters are drawn independently from an ordered alphabet of size $m$, the fluctuations of the shape of the random RSK Young tableaux are investigated, when $n$ and $m$ converge together to infinity. If…
For a subfamily of multiplicative measures on integer partitions we give conditions for properly rescaled associated Young diagrams to converge in probability to a certain deterministic curve named the limit shape of partitions. We provide…
Continuum models for the spatial dynamics of growing cell populations have been widely used to investigate the mechanisms underpinning tissue development and tumour invasion. These models consist of nonlinear partial differential equations…
We introduce a class of cellular automata growth models on the two-dimensional integer lattice with finite cross neighborhoods. These dynamics are determined by a Young diagram $\mathcal Z$ and the radius $\rho$ of the neighborhood, which…
The spread of a particular trait in a cell population often is modelled by an appropriate system of ordinary differential equations describing how the sizes of subpopulations of the cells with the same genome change in time. On the other…
I consider how cell shape and environmental geometry affect the rate of nutrient capture and the consequent maximum growth rate of a cell, focusing on rod-like species like \textit{E.\ coli}. Simple modeling immediately implies that it is…
We compute the limit shapes of the Young diagrams of the minimal difference $p$ partitions and provide a simple physical interpretation for the limit shapes. We also calculate the asymptotic distribution of the largest part of the Young…
Cell size control and homeostasis is a major topic in cell biology yet to be fully understood. Several growth laws like the timer, adder, and sizer were proposed, and mathematical approaches that model cell growth and division were…