The Paquette-Zeitouni law of fractional logarithms for the GUE minor process and the Plancherel growth process

TL;DR

The paper studies laws of fractional logarithms for KPZ-class quantities in the GUE minor process and the Plancherel growth process, connecting eigenvalue fluctuations to Brownian last passage percolation via a distributional identity. It proves almost-sure limsup and liminf constants under $(\log n)^{2/3}$ and $(\log n)^{1/3}$ scalings: for the GUE minor case, $\limsup \frac{(\lambda_n-2\sqrt{n})\,n^{1/6}}{(\log n)^{2/3}}=(1/4)^{2/3}$ and $\liminf \frac{(\lambda_n-2\sqrt{n})\,n^{1/6}}{(\log n)^{1/3}}=-4^{1/3}$; for LIS under Plancherel growth, $\limsup \frac{L_n-2\sqrt{n}}{n^{1/6}(\log n)^{2/3}}=(1/8)^{2/3}$ and $\liminf \frac{L_n-2\sqrt{n}}{n^{1/6}(\log n)^{1/3}}=-2^{1/3}$. The approach relies on sharp Tracy–Widom tail bounds (via BBBK24), decorrelation at sublinear scales, and a Brownian LPP representation (Bar01) to transfer results between eigenvalue problems and growth-process maxima. The work also establishes two general 0-1 laws for these scaling limits and provides an independent proof of prior PZ17 results in this framework. Overall, the results resolve Kalai’s fractional-logarithm questions for both models and deepen understanding of extreme-value scaling in KPZ universality.

Abstract

It is well-known that the largest eigenvalue of an $n\times n$ GUE matrix and the length of a longest increasing subsequence in a uniform random permutation of length $n$, both converge weakly to the GUE Tracy-Widom distribution as $n\to \infty$. We consider the sequences of the largest eigenvalues of the $n\times n$ principal minor of an infinite GUE matrix, and the the lengths of longest increasing subsequences of a growing sequence of random permutations (which, by the RSK bijection corresponds to the top row of the Young diagrams growing according to the Plancherel growth process), and establish laws of fractional logarithms for these. That is, we show that, under a further scaling of $(\log n)^{2/3}$ and $(\log n)^{1/3}$, the $\limsup$ and $\liminf$ respectively of these scaled quantities converge almost surely to explicit non-zero and finite constants. Our results provide complete solutions to two questions raised by Kalai in 2013. We affirm a conjecture of Paquette and Zeitouni (Ann. Probab., 2017), and give a new proof of $\limsup$, due to Paquette and Zeitouni (Ann. Probab., 2017), who provided a partial solution in the case of GUE minor process.

Figures (4)

• Figure 1: To prove Proposition \ref{['lem: liminf lower bound']}, by scaling it is enough to prove \ref{['eq: scaled liminf lower bound']}. To do this we observe that for passage time between $(0,1)$ and $(n,i)$ to be small for some $i$ one of the following events must happen. Either the passage time between $(0,1)$ to $(n-n_{\varepsilon'}, n-n_{\varepsilon'})$ is small or the passage time between $(n-n_{\varepsilon'}, n-n_{\varepsilon'})$ and $(n,i)$ is small. We use the tail estimate from Proposition \ref{['p: lower tail estimates']} to find the desired upper bound for the first event. For each $i$ we again apply the same proposition to get an exponentially small (in $n$) upper bound for the later events. Finally, taking a union bound over $i$ gives us the upper bound in Proposition \ref{['lem: liminf lower bound']}.
• Figure 2: To prove Lemma \ref{['lemma: maximum of point to interval estimates']} we choose the point $(n+n^{1-\varepsilon'},n+n^{1-\varepsilon'})$. We want to find an upper bound for the event that there is $n \leq j \leq n+n^{\frac{2}{3}-\varepsilon'}$ such that the centred and scaled passage time to $(n,j)$ is large. To do this we consider the two events where $D_{n+n^{1-\varepsilon'},n+n^{1-\varepsilon'}}$ is large (this is the event $\mathcal{A}$) and for all $j,$ the minimum passage time between $(n,j)$ and $(n+n^{1-\varepsilon'},n+n^{1-\varepsilon'})$ is not too small (this is the event $\mathcal{B}$). Note that if there is $n \leq j \leq n+n^{\frac{2}{3}-\varepsilon'}$ such that the centred and scaled passage time to $(n,j)$ is large and $\mathcal{B}$ happens then $\mathcal{A}$ also happens. We apply the upper bound for upper tail from Proposition \ref{['p: upper tail estimates']} to find the desired upper bound for $\mathbb{P}(\mathcal{A})$. By applying Proposition \ref{['p: lower tail estimates']} and applying a union bound we can make $\mathbb{P}(\mathcal{B}^c)$ is arbitrarily small.
• Figure 3: To prove Proposition \ref{['p:decorr']} we consider the points $(1,u_1),(1,u_2), \dots$ on the line $x=1$ where $u_k=k^{3+\varepsilon_1}$. Since the parallelogram $P_{u_k}$ (shaded in grey) are disjoint Lemma \ref{['l:indep']} is immediate. Thus to prove the proposition it is sufficient to show Lemma \ref{['l:approx']}. To do this we fix a $k$ and construct the events $\mathcal{A},\mathcal{B}$ and $\mathcal{C}.$ On the event $\mathcal{A}, \Gamma_{1, {u_k}}$ stays within the parallelogram $P_{u_k}$. By Lemma \ref{['lemma: transversal fluctuation']}, (i) $\mathcal{A}^c$ has exponentially small (in $n$) probability. $\mathcal{B}$ is the event that $\Gamma_{1,u_k}$ intersects the line $x=n^{-\frac{\varepsilon_1}{100}}$ inside the interval $J_{u_k}$ (shaded in red). By Lemma \ref{['lemma: transversal fluctuation']}, (ii) $\mathcal{B}^c$ has exponentially small (in $n$) probability. Finally, $\mathcal{C}$ is the event that the maximum (resp. minimum) passage times between $(0,1)$ and $J_{u_k}$ are not too large (resp. not too small). By Proposition \ref{['p: upper tail estimates']} and Proposition \ref{['p: lower tail estimates']} and a union bound over $J_{u_k}$ we conclude that $\mathcal{C}^c$ also has exponentially small (in $n$) probability. Now on the event $\mathcal{A} \cap \mathcal{B} \cap \mathcal{C},$$Y_{u_k}$ and $D_{1,u_{k}}$ are close on the correct scale.
• Figure 4: To prove Lemma \ref{['lemma: transversal fluctuation']}, (ii) we consider a union of events $\mathcal{A}_j$'s on which the distance between $\Gamma_{n,n}(2^jr)$ and $2^jr$ is at least $\ell ((2 \alpha)^jr)^{2/3}$ and the distance between $\Gamma_{n,n}(2^{j+1}r)$ and $2^{j+1}r$ is at most $\ell ((2 \alpha)^{j+1}r)^{2/3}$. Using the first part of the lemma it is enough to consider $\bigcup_{j} \mathcal{A}_j.$ We fix $j$ and show that on the event $\mathcal{A}_j,$ the geodesic between $(0,1)$ and $y_j$ has large global transversal fluctuation from the straight line joining $(0,1)$ and $y_j$. Thus, applying first part of the lemma to each of the $\mathcal{A}_j$'s and applying a union bound gives us the desired upper bound.

Theorems & Definitions (48)

• Theorem 1.1
• Proposition 1.2: Bar01
• Theorem 1.3
• Theorem 1.4
• Proposition 1.5
• Proposition 1.6
• Proposition 1.7
• proof : Proof of Theorem \ref{['t:LPP']}
• Proposition 1.8
• Proposition 1.9