Dependencies

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

A sequence of random variables \(\{ X_n\} \) converges in probability to \(c\) if:

A sequence of random variables \(\{ X_n\} \) converges in probability to zero if:

An edge is a pair of grid points.

For a Gumbel grid and appropriate constants \(C_g, \sigma _g {\gt} 0\):

where \(F_{\text{GUE}}\) is the GUE Tracy-Widom distribution.

A grid path is a list of edges.

A grid point is an element of \(\mathbb {N}^2\).

The Gumbel cumulative distribution function is:

A random field \(Y : \text{Edge} \to \Omega \to \mathbb {R}\) is a Gumbel grid if:

1. The random variables \(\{ Y_e\} _{e \in \text{Edge}}\) are independent, and

2. For each edge \(e\) and \(x \in \mathbb {R}\), \(\mathbb {P}(Y_e \leq x) = F_{\text{Gumbel}}(x)\).

For \(N \geq 1\), define:

For a grid \(E\) of exponential random variables:

Given a weight function \(w : \text{Edge} \to \mathbb {R}\) and endpoints \((m, n)\), the Last Passage Percolation value is:

where the maximum is taken over all valid paths from \((0, 0)\) to \((m, n)\).

For a Gumbel grid \(Y\) and \(N \geq 1\), define:

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

For a Gumbel grid \(Y\), define:

There exists a constant \(D_L {\gt} 0\) such that:

There exists a constant \(D_\ell {\gt} 0\) such that:

An edge \((p, q)\) where \(p = (x, y)\) and \(q = (x', y')\) is up-right if either:

• \(x' = x + 1\) and \(y' = y\) (right step), or

• \(x' = x\) and \(y' = y + 1\) (up step).

A path is valid from point \(p\) to point \(q\) if it is a sequence of connected up-right edges starting at \(p\) and ending at \(q\).

If \(Y_n \xrightarrow {\mathbb {P}} c\) and \(a_n \to 0\), then \(a_n \cdot Y_n \xrightarrow {\mathbb {P}} 0\).

For \(\alpha {\gt} 2/3\):

If \(Y\) is a Gumbel grid, then \(E_e = \exp (-Y_e)\) forms a grid of independent exponential random variables with rate 1.

The Gumbel CDF is continuous on \(\mathbb {R}\).

If \(Y\) is a Gumbel random variable, then for any \(y \in \mathbb {R}\):

If \(Y\) is a Gumbel random variable, then:

If \(Y\) is a Gumbel random variable, then \(\exp (-Y)\) has the exponential distribution with rate 1. Specifically, for \(x \geq 0\):

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 \(x {\gt} 0\).

For any \(m, n \in \mathbb {N}\), there exists at least one valid path from \((0, 0)\) to \((m, n)\).

Let \(\Pi \) be a finite nonempty set, and \(S_A, S_B : \Pi \to \mathbb {R}\) be functions. Suppose \(\pi ^*\) maximizes \(S_A\). Define:

• \(M_A = S_A(\pi ^*)\)

• \(M_{A+B} = \max _{\pi \in \Pi } (S_A(\pi ) + S_B(\pi ))\)

• \(m_B = \min _{\pi \in \Pi } S_B(\pi )\)

• \(M_B = \max _{\pi \in \Pi } S_B(\pi )\)

Then:

For \(N \geq 1\) and \(y \in \mathbb {R}\):

For \(N \geq 1\), a Gumbel grid \(Y\), and \(n \in \mathbb {N}\):

For \(N \geq 1\), a Gumbel grid \(Y\), and \(n \in \mathbb {N}\):

where \(L_{\text{Exp}}(n)\) is computed with weights \(E_e = e^{-Y_e}\).

Assume the properties in Definitions 27, 28, and 29. Let \(N_n = \lfloor n^\alpha \rfloor \) for some \(\alpha {\gt} 0\). For a sequence of Gumbel grids \(Y^{(n)}\):

1. (Convergence for \(\alpha {\gt} 2/3\)) If \(\alpha {\gt} 2/3\), then for any \(r \in \mathbb {R}\):

   \[ \mathbb {P}\left(\frac{T_{\text{Approx}}^{N_n}(n) - C_g \cdot n}{\sigma _g \cdot n^{1/3}} \leq r\right) \to F_{\text{GUE}}(r) \]

   as \(n \to \infty \). That is, the approximate Gumbel LPP converges to the same GUE Tracy-Widom distribution as the exact Gumbel LPP.

2. (Divergence for \(\alpha {\lt} 2/3\)) If \(\alpha {\lt} 2/3\), then the fluctuations diverge:

   \[ \frac{T_{\text{Approx}}^{N_n}(n) - C_g \cdot n}{\sigma _g \cdot n^{1/3}} \xrightarrow {\mathbb {P}} +\infty \]

   as \(n \to \infty \). More precisely, for any \(M {\gt} 0\):

   \[ \mathbb {P}\left(\frac{T_{\text{Approx}}^{N_n}(n) - C_g \cdot n}{\sigma _g \cdot n^{1/3}} {\gt} M\right) \to 1 \]

Thus \(\alpha = 2/3\) represents a sharp threshold: the approximation parameter \(N\) must grow faster than \(n^{2/3}\) for the limiting distribution to remain Tracy-Widom GUE.

Suppose \(X_n \xrightarrow {d} F\) (convergence in distribution to a continuous CDF \(F\)) and \(Y_n \xrightarrow {\mathbb {P}} 0\). Then:

For random variables \(X, Y\) and constants \(r, \varepsilon \):

For random variables \(X, Y\) and constants \(r, \varepsilon \):