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Crystal Growth on Locally Finite Partially Ordered Sets

Tanner J. Reese, Sunder Sethuraman

TL;DR

This work analyzes crystal-growth dynamics on locally finite posets by formulating a last passage percolation problem with independent exponential weights. It derives non-asymptotic, geometry-aware bounds for all moments and the moment generating function of the passage time $\tau_A$, and proves a LLN shape theorem in the monoid setting by exploiting a backward operator $\Delta$ and time-reversed comparisons. The results generalize classical 2D exactly solvable LPP to inhomogeneous and monoid-based contexts, yielding diffusive variance behavior and robust shape bounds via quantities $\ell(A)$, $\kappa(A)$, and $\eta(A)$. The approach hinges on backward-diffusion techniques, path-function constructions, and branching allocations, enabling broad extensions to stochastically monotone weight families. The findings have potential implications for understanding universal fluctuation scales and growth dynamics in generalized directed-percolation-like systems.

Abstract

We consider a Markovian growth process on a partially ordered set $Λ$, equivalent to last passage percolation (LPP) with independent (not necessarily identical) exponentially distributed weights on the elements of $Λ$. Such a process includes inhomogeneous exponential LPP on the Euclidean lattice $\mathbb{N}_0^d$. We give non-asymptotic bounds on the mean and variance, as well as higher, central, and exponential moments of the passage time $τ_A$ to grow any set $A \subseteq Λ$ in terms of characteristics of $A$. We also give a limit shape theorem when $Λ$ is equipped with a monoid structure. Methods involve making use of the backward equation associated to the Markovian evolution and comparison inequalities with respect to the time-reversed generator.

Crystal Growth on Locally Finite Partially Ordered Sets

TL;DR

This work analyzes crystal-growth dynamics on locally finite posets by formulating a last passage percolation problem with independent exponential weights. It derives non-asymptotic, geometry-aware bounds for all moments and the moment generating function of the passage time , and proves a LLN shape theorem in the monoid setting by exploiting a backward operator and time-reversed comparisons. The results generalize classical 2D exactly solvable LPP to inhomogeneous and monoid-based contexts, yielding diffusive variance behavior and robust shape bounds via quantities , , and . The approach hinges on backward-diffusion techniques, path-function constructions, and branching allocations, enabling broad extensions to stochastically monotone weight families. The findings have potential implications for understanding universal fluctuation scales and growth dynamics in generalized directed-percolation-like systems.

Abstract

We consider a Markovian growth process on a partially ordered set , equivalent to last passage percolation (LPP) with independent (not necessarily identical) exponentially distributed weights on the elements of . Such a process includes inhomogeneous exponential LPP on the Euclidean lattice . We give non-asymptotic bounds on the mean and variance, as well as higher, central, and exponential moments of the passage time to grow any set in terms of characteristics of . We also give a limit shape theorem when is equipped with a monoid structure. Methods involve making use of the backward equation associated to the Markovian evolution and comparison inequalities with respect to the time-reversed generator.
Paper Structure (21 sections, 30 theorems, 125 equations, 5 figures)

This paper contains 21 sections, 30 theorems, 125 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Discussion of Previous Literature
  3. Sketch of Results and Methods
  4. Plan of the Article
  5. Crystal Growth Markov Process
  6. Well-definedness of Functions on $L(\Lambda)$
  7. Relationship to Last Passage Percolation
  8. Results
  9. Upper Bounds on $\mathrm{Var}_{}\left(\tau_A\right)$ and Higher Moments in Terms of $\mathbb{E}_{}\left[\tau_A\right]$
  10. Lower Bound on $\mathrm{Var}_{}\left(\tau_A\right)$
  11. Bounds on $\mathbb{E}_{}\left[\tau_A\right]$
  12. Bounds on the Moment Generating Function of $\tau_A$
  13. Partial Orders on Monoids
  14. Extension to Stochastically Monotone Weights
  15. Properties of the Backward Operator $\Delta$
  16. ...and 6 more sections

Key Result

Lemma 2.2

For any $A \in L(\Lambda)$ and $t \geq 0$, where $A \mapsto \mathbb{P}_{}\left(\tau_A \leq t\right)$ is treated as a function on $L(\Lambda)$.

Figures (5)

  • Figure 1: The positive cone $C$ in $\overline{\Lambda} = \mathbb{Z}^2$
  • Figure 2: $\Lambda$ is a tree order with two elements immediately above each element
  • Figure 3: The maximal elements, $\mathcal{M}(A)$, and the growth elements, $\mathcal{M}^*(A)$, for a lower set $A \subset \Lambda= \mathbb{N}_0^2$.
  • Figure 4: Three maximal paths in the lower set $A \subseteq \Lambda = \mathbb{N}_0^2$
  • Figure 5: The lower set $A$ in the poset $\mathbb{N}_0 \times \{1, \ldots, 5\}$ where the ordering is determined by $\mathbb{N}_0$ (up is greater) and $(n, a)$ is incomparable to $(n, b)$ whenever $a \neq b$

Theorems & Definitions (73)

  • Example 1.1
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Proposition 2.4
  • Theorem 3.1
  • Proposition 3.2
  • Corollary 3.3
  • ...and 63 more