Technical Summary: Large Deviations at the Edge for 1D Gases and Tridiagonal Random Matrices at High Temperature

1. Problem Setting

This paper investigates the large deviation (LD) behavior of the largest particle, $x_{\max} = \max_{1 \le i \le N} x_i$, in a one-dimensional gas of $N$ confined particles subject to a two-body repulsive interaction. The system is governed by the Gibbs probability measure:
$$dP_N(x) = \frac{1}{Z_N} \prod_{i<j} e^{-2\alpha \frac{1}{N} g(x_i - x_j)} \prod_{i=1}^N e^{-V(x_i)} dx_1 \dots dx_N$$
where the interaction kernel $g$ is either logarithmic ($g(x) = -\log|x|$) or Riesz ($g(x) = |x|^{-s}$ with $0 < s < 1$).

The study focuses on the high-temperature regime, defined by the scaling of the inverse temperature $\beta_N = 2\alpha/N$ for a fixed $\alpha > 0$. In this regime, the empirical measure converges to an equilibrium measure $\mu_{eq}$ supported on the entire real line (unlike the compact support in the fixed $\beta$ regime). Consequently, the typical scale of the largest particle grows as $x_{\max} \sim (\theta^{-1} \log N)^{1/d}$, where $V(x) \sim \theta |x|^d$.

The primary objective is to quantify the probability of deviations of $x_{\max}$ from this typical behavior as $N \to \infty$. Specifically, the authors seek to establish a Large Deviation Principle (LDP) and moderate deviation estimates for the rescaled variable $\hat{x}_{\max} = x_{\max} / (\theta^{-1} \log N)^{1/d}$.

Additionally, the paper analyzes symmetric tridiagonal random matrices with independent entries possessing Gaussian tails. This analysis is motivated by the connection between the log-gas (with quadratic potential) and the eigenvalues of tridiagonal matrices (the Dumitriu-Edelman representation), allowing the authors to interpret LD events in terms of matrix entries.

2. Methodology

2.1. Large Deviation Analysis for Gases

The proof strategy for the gas models (Theorems 1.2 and 1.3) relies on a combination of upper and lower bound techniques:

• Upper Bounds (Right Deviations): For deviations where $x_{\max}$ is larger than typical ($x > 1$), the authors utilize a Wegner estimate (marginal bound) on the first particle derived from previous work on local statistics. By applying a union bound and exchangeability, they bound the probability that any particle exceeds a threshold.
• Upper Bounds (Left Deviations): For deviations where $x_{\max}$ is smaller than typical ($x < 1$), the strategy involves comparing the interacting system to a system of independent particles distributed according to the equilibrium measure $\mu_{eq}^{\otimes N}$. Using energy estimates and a regularization of the empirical measure, the authors establish a pointwise bound on the density $p_N(x)$:
  $$p_N(x) \le e^{R_N} \prod_{i=1}^N \rho_{eq}(x_i)$$
  where $R_N$ is a remainder term that is negligible compared to the exponential scale of the large deviations. This reduces the problem to estimating the probability for independent particles, which is more tractable.
• Lower Bounds: The lower bound is established by restricting the configuration space to events where exactly one particle deviates significantly while the remaining $N-1$ particles stay within the typical support. The convergence of local statistics at the edge to an inhomogeneous Poisson point process (established in [Lam21]) is crucial here, allowing the authors to bound the probability of a single particle deviating from below.
• Moderate Deviations: The paper extends these results to the moderate deviation regime, where the deviation scale is smaller than the large deviation scale but larger than the typical fluctuations (Gumbel regime).

2.2. Matrix Interpretation and Truncation

For the tridiagonal matrix models (Theorem 1.10), the authors employ a truncation procedure similar to techniques used for sparse matrices and matrices with heavy tails:

• Truncation: The matrix $T$ is decomposed into a truncated matrix $T^\varepsilon$ (containing only entries larger than $\varepsilon \sqrt{2 \log N}$) and a remainder $S^\varepsilon$.
• Exponential Equivalence: It is shown that the top eigenvalues of $T$ and $T^\varepsilon$ are exponentially equivalent; the probability that they differ by more than a small $\delta$ decays super-exponentially.
• Block Diagonalization: Due to the Gaussian tail assumption, the truncated matrix $T^\varepsilon$ is sparse. With high probability, the off-diagonal entries that remain are few, causing the matrix to decompose into a block diagonal structure with blocks of finite size (independent of $N$).
• Norm Bounds: The top eigenvalue of a finite block is bounded by its Euclidean norm. Since the entries have Gaussian tails, the norm of a finite-dimensional vector of such entries also exhibits Gaussian tail behavior. This allows the derivation of the LD rate function.

3. Key Contributions and Results

3.1. Large Deviation Principle for the Log-Gas

Theorem 1.2 establishes a full LDP for the rescaled largest particle $\hat{x}_{\max}$ in the log-gas case ($g = -\log$). The sequence satisfies an LDP with speed $\log N$ and the good rate function:
$$I_d(x) = \begin{cases} x^d - 1 & \text{if } x \ge 1 \\ +\infty & \text{if } x < 1 \end{cases}$$
This result implies that the rate function for the interacting log-gas in the high-temperature regime is identical to that of a system of independent and identically distributed (iid) particles. The correlations introduced by the interaction do not affect the large deviation cost of the largest particle in this regime.

3.2. Estimates for the Riesz Gas

Theorem 1.3 provides LD estimates for the Riesz gas ($g = |x|^{-s}$). While a full LDP is not proven for all $x < 1$, the authors establish:

• An upper bound for $x > 1$: $\limsup \frac{1}{\log N} \log P(\hat{x}_{\max} > x) \le -I_d(x)$.
• A lower bound for $x > 1$ and specific regions of $x < 1$.
• The rate function is again $I_d(x)$, suggesting the same universality as the log case, though the proof for the left tail ($x < 1$) is more delicate due to the interaction range.

3.3. Refined Bounds and Moderate Deviations

• Proposition 1.5: Provides a refined upper bound for left deviations ($x < 1$), showing that the probability of the largest particle being significantly smaller than typical decays faster than any power of $N$ (specifically, the log-probability scales with $\log \log N$). This indicates that left deviations require a mesoscopic number of particles to shift, making them much rarer than right deviations.
• Proposition 1.7: Establishes moderate deviation estimates for the regime where the deviation is of order $(\log N)^{-\gamma}$. The authors show that the probability of such deviations decays as $\exp(-x^d (\log N)^{1-\gamma-d^{-1}})$.

3.4. Tridiagonal Matrices and the "Few Entries" Mechanism

Theorem 1.10 establishes an LDP for the top eigenvalue of symmetric tridiagonal matrices with independent entries having uniform Gaussian tails. The rate function is $I_2(x)$ (corresponding to $d=2$ in the potential $V(x) \sim x^2$).

• Mechanism: The paper demonstrates that large deviations of the top eigenvalue are driven by a small number of entries taking abnormally large values.
• Recovery of Log-Gas: By specializing the matrix entries to the Dumitriu-Edelman model (where diagonal entries are Gaussian and off-diagonal entries follow a $\chi$-distribution), the authors recover the LDP for the log-gas (Theorem 1.2), providing a matrix-theoretic interpretation of the gas results.

4. Significance and Claims

The paper claims to extend the understanding of large deviations in random matrix theory and statistical mechanics to the high-temperature regime ($\beta_N \to 0$).

• Universality of the Rate Function: A central claim is that in the high-temperature limit, the large deviation rate function for the largest particle in log and Riesz gases coincides with that of iid particles. This contrasts with the fixed $\beta > 0$ regime, where interactions significantly alter the rate function (e.g., in the classical $\beta$-ensembles, the rate function involves a different energy term).
• Nature of Rare Events: The work elucidates the mechanism of rare events. For right deviations (large $x_{\max}$), the event is driven by a single particle escaping the bulk. For left deviations (small $x_{\max}$), the event requires a collective shift of a mesoscopic number of particles, making it exponentially more suppressed.
• Matrix-Gas Connection: The paper bridges the gap between interacting particle systems and random matrix theory by showing that the LD behavior of the top eigenvalue in specific tridiagonal models is entirely determined by the tail behavior of the individual matrix entries, rather than complex spectral correlations. This supports the intuition that in the high-temperature (weakly correlated) limit, the system behaves locally like a collection of independent variables.

The authors position their results as a continuation of works on fixed $\beta$ ensembles (e.g., [BADG01], [Pak20]) and high-temperature empirical measure convergence (e.g., [GZ19]), filling the gap regarding the edge statistics and large deviations in the $\beta_N \sim 1/N$ regime.