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Large-population asymptotics for the maximum of diffusive particles with mean-field interaction in the noises

Nikolaos Kolliopoulos, David Sanchez, Amy Xiao

TL;DR

The work addresses the large-$N$ behavior of the normalized maximum of $N$ diffusive particles with mean-field noise interactions. It introduces a universal time-change that relocates noise interaction into the drift, reducing the problem to a McKean–Vlasov-type EVT setting where the single-particle law is Gaussian. In a simple Gaussian regime, the maximum converges to the standard Gumbel distribution $P(x)=e^{-e^{-x}}$, and under suitable regularity and moment conditions, the interacting system inherits this limit (possibly with stochastic normalizers that can be promoted to deterministic ones under stronger convergence). The paper also reports numerical simulations on a bank-network model, showing that mean-field noise accelerates convergence to the Gumbel law compared with i.i.d. McKean–Vlasov particles, with implications for modeling extreme events and systemic risk in large interacting systems.

Abstract

We study the $N \to \infty$ limit of the normalized largest component in some systems of $N$ diffusive particles with mean-field interaction. By applying a universal time change, the interaction in noises is transferred to the drift terms, and the asymptotic behavior of the maximum becomes well-understood due to existing results in the literature. We expect that the normalized maximum in the original setting has the same limiting distribution as that of i.i.d copies of a solution to the corresponding McKean-Vlasov SDE and we present some results and numerical simulations that support this conjecture.

Large-population asymptotics for the maximum of diffusive particles with mean-field interaction in the noises

TL;DR

The work addresses the large- behavior of the normalized maximum of diffusive particles with mean-field noise interactions. It introduces a universal time-change that relocates noise interaction into the drift, reducing the problem to a McKean–Vlasov-type EVT setting where the single-particle law is Gaussian. In a simple Gaussian regime, the maximum converges to the standard Gumbel distribution , and under suitable regularity and moment conditions, the interacting system inherits this limit (possibly with stochastic normalizers that can be promoted to deterministic ones under stronger convergence). The paper also reports numerical simulations on a bank-network model, showing that mean-field noise accelerates convergence to the Gumbel law compared with i.i.d. McKean–Vlasov particles, with implications for modeling extreme events and systemic risk in large interacting systems.

Abstract

We study the limit of the normalized largest component in some systems of diffusive particles with mean-field interaction. By applying a universal time change, the interaction in noises is transferred to the drift terms, and the asymptotic behavior of the maximum becomes well-understood due to existing results in the literature. We expect that the normalized maximum in the original setting has the same limiting distribution as that of i.i.d copies of a solution to the corresponding McKean-Vlasov SDE and we present some results and numerical simulations that support this conjecture.
Paper Structure (3 sections, 2 theorems, 41 equations, 3 figures)

This paper contains 3 sections, 2 theorems, 41 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Elimination of noise interaction via time change: a simple Gaussian setting
  3. A model for a network of banks: numerical simulations

Key Result

Theorem 2.3

Consider the transformation $t \mapsto \tau_N(t)$ with Under Assumption ass1 we have that $\tau_N(t) \to \tau(t) := \int^t_0 \sigma^2\left(\mathbb{E}\left[g^{\sigma}(X_s)\right]\right)ds$ as $N \to \infty$, for any $t \geq 0$. Moreover, for each fixed $x \in \mathbb{R}$ and $T > 0$, the function converges weakly in $L^2([0, T])$ to $P(x) = e^{-e^{-x}}$ as $N \to \infty$. The latter means that f

Figures (3)

  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (5)

  • Remark 2.2
  • Theorem 2.3
  • Proposition 2.4
  • proof : Proof of Theorem \ref{['maintheorem1']}
  • proof : Proof of Proposition \ref{['maintheorem2']}