Approximate Gumbel Last Passage Percolation

4 The Coupling Construction

4.1 Approximate Gumbel Distribution

Definition 15 Approximate Gumbel CDF
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For \(N \geq 1\), the \(N\)-approximate Gumbel CDF is:

\[ F_N(x) = \begin{cases} \left(1 - \frac{e^{-x}}{N}\right)^N & \text{if } x {\gt} -\log N \\ 0 & \text{otherwise} \end{cases} \]

4.2 The Coupling Function

Definition 16 Coupling Function \(h_N\)
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For \(N \geq 1\), define:

\[ h_N(x) = -\log \left(N \cdot \left(1 - e^{-e^{-x}/N}\right)\right) - x \]
Lemma 17 Convexity and Bounds for \(h_N\)
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For \(N \geq 1\), the function \(h_N : \mathbb {R} \to \mathbb {R}\) satisfies:

  1. \(h_N\) is convex on \(\mathbb {R}\),

  2. \(0 {\lt} h_N(x) \leq \frac{e^{-x}}{N}\) for all \(x \in \mathbb {R}\),

  3. \(\frac{e^{-x}}{3N} \leq h_N(x)\) for all \(x {\gt} 0\).

Proof ▶

The proof uses calculus to verify convexity by showing the second derivative is non-negative. The upper bound follows from Taylor expansion of the exponential and logarithm. The lower bound for \(x {\gt} 0\) uses the inequality \(1 - e^{-t} \geq t - \frac{t^2}{2} + \frac{t^3}{6}\) for \(t \geq 0\).

Theorem 18 Coupling Identity
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For \(N \geq 1\) and \(y \in \mathbb {R}\):

\[ F_{\text{Gumbel}}(y) = F_N(h_N(y) + y) \]
Proof ▶

Direct calculation shows both sides equal \(\exp (-e^{-y})\).

4.3 LPP Definitions

Definition 19 Gumbel LPP
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For a Gumbel grid \(Y\), define:

\[ T_{\text{Gumbel}}(n) = \text{LPP}_Y(n, n) \]
Definition 20 Approximate Gumbel LPP
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For a Gumbel grid \(Y\) and \(N \geq 1\), define:

\[ T_{\text{Approx}}^N(n) = \text{LPP}_{Y + h_N(Y)}(n, n) \]

where the weights are \(w_e = Y_e + h_N(Y_e)\) for each edge \(e\).

Definition 21 Exponential LPP
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For a grid \(E\) of exponential random variables:

\[ L_{\text{Exp}}(n) = \text{LPP}_E(n, n) \]