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Stochastic domination and lifts of random variables in percolation theory

Sébastien Martineau, Rémy Poudevigne, Paul Rax

Abstract

Consider some matrix waiting for its coefficients to be written. For each column, sample independently a Bernoulli random variable of some parameter $p$. Seeing all this and possibly using extra randomness, Alice then chooses one spot in each column, in any way she wants. When the Bernoulli random variable of some column is equal to 1, the number 1 is written in the chosen spot. When the Bernoulli random variable of a column is 0, nothing is done on this column. We prove that, using extra randomness, it is possible for Bob to fill the empty entries with well chosen 0's and 1's so that the entries of the matrix are independent Bernoulli random variables of parameter $p$. We investigate various generalisations and variations of this problem, and use this result to revisit and generalise (nonstrict) monotonicity of the percolation threshold $p_c$ with respect to a form of graph-quotienting, namely fibrations. We also use this result to revisit the BK inequality. In a second part, which is independent of the first one, we revisit strict monotonicity of $p_c$ with respect to fibrations, a result that naturally requires more assumptions than its nonstrict counterpart. We reprove the bond-percolation case of the result of Martineau--Severo without resorting to essential enhancements, using couplings instead.

Stochastic domination and lifts of random variables in percolation theory

Abstract

Consider some matrix waiting for its coefficients to be written. For each column, sample independently a Bernoulli random variable of some parameter . Seeing all this and possibly using extra randomness, Alice then chooses one spot in each column, in any way she wants. When the Bernoulli random variable of some column is equal to 1, the number 1 is written in the chosen spot. When the Bernoulli random variable of a column is 0, nothing is done on this column. We prove that, using extra randomness, it is possible for Bob to fill the empty entries with well chosen 0's and 1's so that the entries of the matrix are independent Bernoulli random variables of parameter . We investigate various generalisations and variations of this problem, and use this result to revisit and generalise (nonstrict) monotonicity of the percolation threshold with respect to a form of graph-quotienting, namely fibrations. We also use this result to revisit the BK inequality. In a second part, which is independent of the first one, we revisit strict monotonicity of with respect to fibrations, a result that naturally requires more assumptions than its nonstrict counterpart. We reprove the bond-percolation case of the result of Martineau--Severo without resorting to essential enhancements, using couplings instead.

Paper Structure

This paper contains 29 sections, 21 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Stochastic domination of lifts
  3. General background regarding percolation theory
  4. Revisiting a percolation result via Theorem \ref{['lakon']}
  5. Other results we revisit
  6. Main theorem
  7. Statement
  8. The one-column case
  9. From one column to the finite case
  10. From the finite setup to the general case
  11. Proofs of the corollaries
  12. Proof of Corollary \ref{['coro:indep']}
  13. Proof of Corollary \ref{['coro:exchange']}
  14. Variations on Theorem \ref{['main']}
  15. Lifting several variables per column
  16. ...and 14 more sections

Figures (4)

  • Figure 1: Illustration of the notation of Theorem \ref{['lakon']}.
  • Figure 2: This table justifies that our construction satisfies Assumption $\mathbf{C}_\mu^\rho$.
  • Figure 3: Going from $\mathscr{S}_0$ to $\mathscr{S}$ and defining the cells.
  • Figure 4: Assuming that the $X$'s and $Y$'s are 0 in the neighbouring cells, this is a situation where $u$ belongs to $K^+_{p,s}(v)$ but $v$ does not belong to $K^+_{p,s}(u)$.

Theorems & Definitions (10)

  • proof : Sketch of proof of Theorem \ref{['thm:bs']} using Theorem \ref{['lakon']}
  • proof
  • proof
  • proof
  • proof
  • proof : Proof of Proposition \ref{['finite']}
  • proof : Proof of Theorem \ref{['thm:multilift']}
  • proof
  • proof : Proof of the BK inequality given our results
  • proof