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Infinite-Particle Determinantal Dynamics from Brownian Motion on $\mathrm{GL}_N(\mathbb{C})$

This paper establishes that the edge scaling limit of the logarithms of squared singular values for multiplicative Brownian motion on $\mathrm{GL}_N(\mathbb{C})$ yields infinite-particle determinantal dynamics with an explicit double-contour correlation kernel, where the dependence on arbitrary initial conditions (including coinciding coordinates) is encoded by Laguerre–Pólya entire functions.

Original authors: Zahra Sadat Mirsajjadi

Published 2026-09-29

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CC BY 4.0

Original authors: Zahra Sadat Mirsajjadi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of modern mathematics, there is a field dedicated to understanding how randomness shapes order. Imagine a crowd of people moving through a city; if they move completely at random, they might bump into each other, but if they are governed by specific rules that prevent them from colliding, their collective movement creates a surprising, structured pattern. This is the world of random matrix theory, where scientists study large grids of numbers that change over time according to the laws of probability. A central question in this field is how these complex, interacting systems behave when they become infinitely large. Do they settle into a predictable rhythm, or do they remain chaotic? Understanding these limits helps physicists and mathematicians describe everything from the energy levels of atoms to the behavior of massive networks, revealing that deep beneath the noise of randomness, there often lies a hidden, elegant geometry.

A recent study by Zahra Sadat Mirsajjadi tackles a specific and challenging version of this problem involving a type of random motion that happens on a group of complex matrices, which are essentially grids of numbers used to describe transformations in space. The researcher focused on what happens to the "squared singular values" of these matrices as they evolve. In simpler terms, these values act like a set of coordinates that track the size and shape of the matrix as it changes. The study looked at the logarithms of these values, which effectively stretches out the data to make the patterns easier to see, and examined how they behave when the number of particles in the system grows to infinity. The key innovation of this work is that it does not assume the system starts from a simple, perfect state. Instead, it investigates what happens when the system begins in any possible configuration, including messy ones where some particles might start in the exact same spot or where the initial data is incomplete.

The paper proves that even with these complicated and varied starting points, the system eventually settles into a highly organized state known as a determinantal point process. This is a specific mathematical structure where the probability of finding particles at certain locations can be calculated using a single, unifying formula. The researcher derived an explicit recipe, written as a double-contour integral, that allows anyone to calculate the correlation between particles at different times and places. This formula is powerful because it encodes the entire history of the initial state, not just the positions of the particles, but also hidden information about the system's starting conditions. This hidden information is captured by a special type of mathematical function, which acts as a fingerprint for the initial setup. The study shows that two systems starting with identical particle positions but different hidden fingerprints will evolve into different patterns, proving that the initial state contains more information than just the visible coordinates.

One of the most significant findings is that the system behaves in a way that is robust even when particles start on top of each other. In many physical models, having particles start at the same location causes the mathematics to break down or become undefined. However, this research demonstrates that the system naturally resolves these collisions instantly. The particles separate immediately and begin to move in a strictly ordered fashion, never crossing paths again. The author provides a new way to describe this motion using a system of equations that includes a "singular interaction," a term for a force that becomes infinitely strong when particles get too close. By treating this force as a limit that is approached over time, the study constructs a valid description of the system's evolution from any starting point, including the most degenerate cases.

The work also confirms that the system possesses a property called the Gibbs resampling property. This means that if you were to freeze the system at a specific moment and look at the path of a single particle, its future movement would depend on the positions of its neighbors in a very specific, probabilistic way, similar to how a rubber band stretches between fixed points. This property holds true for every possible initial condition, extending previous results that were limited to simpler starting scenarios. By establishing these rules, the paper completes the mathematical description of this infinite-particle system, showing that it is not only predictable but also possesses a deep, underlying structure that persists regardless of how chaotic the beginning might be. The findings provide a rigorous foundation for understanding how complex, interacting systems self-organize, offering a clear window into the deterministic laws that govern the edge of randomness.

Technical Summary: Infinite-Particle Determinantal Dynamics from Brownian Motion on $GL_N(\mathbb{C})$

Problem Statement
This paper investigates the edge scaling limit of the logarithms of the squared singular values of multiplicative Brownian motion on the general linear group $GL_N(\mathbb{C})$ with general initial conditions. While previous work (specifically [9]) established the path-space convergence of these dynamics to an infinite-dimensional Feller process and derived a stochastic partial differential equation (SPDE) for the reverse characteristic polynomial, the explicit correlation structure of the limiting process for arbitrary initial states remained undetermined. The central problem is to establish whether this limiting process possesses a determinantal structure for general initial configurations (including those with coinciding or degenerate coordinates) and to derive explicit formulas for its fixed-time and extended multitime correlation kernels.

Methodology
The author employs a rigorous asymptotic analysis of finite-dimensional correlation kernels derived from non-colliding Brownian motions with drift. The methodology proceeds in several stages:

Finite-Dimensional Representation: The squared singular values of the matrix process are related to non-colliding Brownian motions with equally spaced drifts. The author first derives double-contour integral formulas for the fixed-time and multitime correlation kernels of these finite- $N$ systems, utilizing the Eynard–Mehta theorem and residue calculus on non-colliding Brownian bridges with arithmetic terminal points.
Edge Scaling and Conjugation: The finite- $N$ kernels are subjected to edge scaling ( $x \mapsto N^{-1}x$ and logarithmic transformation). A critical technical step involves conjugating the kernels by specific exponential factors to cancel divergent terms arising from the scaling limit (specifically factors of the form $e^{N(w-z)}$ ).
Contour Deformation: The integration contours, which initially depend on $N$ to enclose the finite set of poles, are deformed to fixed contours (a vertical line $\Gamma$ and a half-strip $\Lambda$ ). This allows for the application of the dominated convergence theorem.
Limit Passage: The author proves that the sequence of finite- $N$ kernels converges locally uniformly to a limiting kernel. The dependence on the initial state is encoded through the convergence of associated entire functions (specifically, the Laguerre–Pólya class) defined by the initial configuration.
Stochastic Analysis: To extend the Gibbs resampling property and the infinite-dimensional stochastic differential equation (ISDE) description to arbitrary initial states (including those with coinciding coordinates), the author utilizes the determinantal structure to prove strict ordering of particles at positive times. They then construct solutions to the singular ISDE via an improper time integral limit.

Key Contributions and Results

Determinantal Structure for General Initial Conditions: The paper proves that for every initial state in the enhanced state space $\Upsilon$ (which includes information beyond just particle coordinates), the limiting edge-scaled logarithmic dynamics form a determinantal point process. This extends previous results which were largely restricted to the identity initial condition or strictly ordered configurations.
Explicit Correlation Kernels: The author derives explicit double-contour integral formulas for both the fixed-time kernel (Theorem 1.3) and the extended multitime kernel (Theorem 1.4).
- The kernels depend on the initial state $\varpi = (\mathbf{x}, \gamma)$ through an associated Laguerre–Pólya entire function $\varphi_\varpi(z)$ .
- The formula for the extended kernel includes a transition term $P_{s,t}(u,v)$ and a double integral involving $\varphi_\varpi(e^{-w})/\varphi_\varpi(e^{-z})$ .
Encoding of Initial Data: A significant finding is that the initial state is encoded entirely by the Laguerre–Pólya function. Crucially, distinct initial states with identical particle coordinates but different values of the parameter $\gamma$ (representing "hidden" information in the enhanced state space) yield different correlation kernels. This implies that identical particle configurations can give rise to different laws for the dynamics.
Extension to Degenerate Configurations: The paper establishes that the limiting dynamics satisfy the Brownian Gibbs property and the associated ISDE for arbitrary initial configurations, including those with coinciding coordinates.
- Theorem 1.6 and Corollary 1.7 provide an ISDE formulation where the singular interaction at time zero is interpreted as an almost surely improper time integral.
- It is shown that even if active particles start at coinciding coordinates, they separate instantaneously and remain strictly ordered for all $t > 0$ .
Alternative Representations: For the specific case of the identity initial condition, the derived formulas provide alternative contour-integral representations for the correlation kernels previously studied in [1].

Significance and Claims
The paper claims to complete the probabilistic description of the limiting dynamics for multiplicative Brownian motion on $GL_N(\mathbb{C})$ . By establishing the determinantal structure and explicit kernels for the full enhanced state space, the work bridges the gap between the path-space convergence results of [9] and the explicit correlation formulas available for specific initial conditions.

The author notes that the construction of solutions to the infinite-dimensional singular log-interacting SDE from all admissible deterministic initial data, particularly those with coinciding coordinates, is novel. The results provide a framework for understanding non-equilibrium infinite-particle systems where the initial configuration is not fully determined by the particle positions alone, but also by an associated entire function. The paper concludes by suggesting that these explicit kernels may facilitate future work on fixed-time rigidity estimates and the investigation of uniqueness for the dynamics on rigid-path spaces, though these are left as open questions for future research.

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