{
  "abstract": "We study a class of interacting particle systems on $\\mathbb{R}$ with two types. Particles evolve by independent jumps sampled from a fixed distribution, with type-dependent jump rates $v_+$, $v_-$ and stochastic type switching driven by non-local order-based interactions. The switching rates depend on the empirical distribution through the proportion of opposite-type particles located ahead, leading to a nonlinear and discontinuous dependence on the empirical measure outside the standard Lipschitz McKean-Vlasov framework. Our first main result is a law of large numbers for the empirical measure process: we prove convergence, along subsequences, to a deterministic measure-valued process characterized by a McKean-Vlasov equation. The proof combines tightness in Wasserstein space with a martingale characterization of limit points. A uniqueness argument based on a Kolmogorov-Smirnov-type distance adapted to the ordering structure yields convergence of the full empirical measure sequence and, in turn, propagation of chaos on finite time intervals. We then study the long-time behavior of the limiting dynamics. Because the system has persistent drift, invariant distributions do not arise; instead, we analyze traveling waves, corresponding to stationary profiles in a moving frame. For exponential jump distributions, the associated non-local integro-differential system admits a local description. In the regime $v_+>v_-=0$, this further reduces to a coupled system of non-linear ODEs, allowing a phase-plane analysis that yields a traveling wave as a heteroclinic orbit connecting two equilibria. We also identify the wave speed and mass partition, and derive tail asymptotics by spectral analysis of the linearized system.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.11279",
  "author_total": 2,
  "authors": [
    {
      "name": "Sayan Banerjee"
    },
    {
      "name": "Andrew Nguyen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d7vn1fbdvmn7q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.11279",
  "fulltext_url": null,
  "id": "d7vn1fbdvmn7q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.11279",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7161048552",
  "pdf_url": "https://arxiv.org/pdf/2605.11279",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d7vn1fbdvmn7q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2605.11279",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.11279"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Flocking with Multiple Types: Competition, Fluid Limits and Traveling Waves",
  "title_text": "Flocking with Multiple Types: Competition, Fluid Limits and Traveling Waves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/d7vn1fbdvmn7q",
  "venue": "arXiv",
  "versions": []
}