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Fluid limit and gelation in the frozen Erdős-Rényi random graph

Bénédicte Haas, Vincent Viau

TL;DR

This work analyzes a p-frozen variant of the Erdős-Rényi dynamic graph, where unicyclic components are frozen to inhibit giant components. The authors establish a fluid limit for key statistics (gel size, discarded edges, and forest parameters) by adapting Wormald's differential equation method to a two-dimensional state, $(G_{p,n},D_{p,n})$, and show convergence to a deterministic system defined by the gel function $g_p$ and its companion $d_p$. They connect the frozen model to uniform random forests via the free-forest property, enabling precise asymptotics for the jumps and component sizes, as well as for the number of trees of fixed size and their threshold behavior. A continuous Poissonized version is introduced to obtain exact tractable results, which are then de-Poissonized to yield absorption-time results: the total gelation time scales like $A_{p,n}\sim \frac{n\ln n}{2p}$ with a Gumbel-type limit around critical thresholds. Overall, the paper provides a comprehensive rigorous description of gelation dynamics, critical-window behaviour, and asymptotics of the time to full freezing, highlighting the role of the slowdown parameter $p$ in shaping the dynamical phase and component structure. The results extend and corroborate physical predictions and connect to classical ER theory through the gel and forest decompositions.

Abstract

The frozen Erdős-Rényi random graph is a variant of the standard dynamical Erdős-Rényi random graph that prevents the creation of the giant component by freezing the evolution of connected components with a unique cycle. The formation of multicyclic components is forbidden, and the growth of components with a unique cycle is slowed down, depending on a parameter $p\in [0,1]$ that quantifies the slowdown. At the time when all connected components of the graph have a (necessary unique) cycle, the graph is entirely frozen and the process stops. In this paper we study the fluid limit of the main statistics of this process, that is their functional convergence as the number of vertices of the graph becomes large and after a proper rescaling, to the solution of a system of differential equations. Our proofs are based on an adaption of Wormald's differential equation method. We also obtain, as a main application, a precise description of the asymptotic behavior of the first time when the graph is entirely frozen.

Fluid limit and gelation in the frozen Erdős-Rényi random graph

TL;DR

This work analyzes a p-frozen variant of the Erdős-Rényi dynamic graph, where unicyclic components are frozen to inhibit giant components. The authors establish a fluid limit for key statistics (gel size, discarded edges, and forest parameters) by adapting Wormald's differential equation method to a two-dimensional state, , and show convergence to a deterministic system defined by the gel function and its companion . They connect the frozen model to uniform random forests via the free-forest property, enabling precise asymptotics for the jumps and component sizes, as well as for the number of trees of fixed size and their threshold behavior. A continuous Poissonized version is introduced to obtain exact tractable results, which are then de-Poissonized to yield absorption-time results: the total gelation time scales like with a Gumbel-type limit around critical thresholds. Overall, the paper provides a comprehensive rigorous description of gelation dynamics, critical-window behaviour, and asymptotics of the time to full freezing, highlighting the role of the slowdown parameter in shaping the dynamical phase and component structure. The results extend and corroborate physical predictions and connect to classical ER theory through the gel and forest decompositions.

Abstract

The frozen Erdős-Rényi random graph is a variant of the standard dynamical Erdős-Rényi random graph that prevents the creation of the giant component by freezing the evolution of connected components with a unique cycle. The formation of multicyclic components is forbidden, and the growth of components with a unique cycle is slowed down, depending on a parameter that quantifies the slowdown. At the time when all connected components of the graph have a (necessary unique) cycle, the graph is entirely frozen and the process stops. In this paper we study the fluid limit of the main statistics of this process, that is their functional convergence as the number of vertices of the graph becomes large and after a proper rescaling, to the solution of a system of differential equations. Our proofs are based on an adaption of Wormald's differential equation method. We also obtain, as a main application, a precise description of the asymptotic behavior of the first time when the graph is entirely frozen.

Paper Structure

This paper contains 43 sections, 48 theorems, 287 equations, 2 figures.

Table of Contents

  1. Introduction and main results
  2. Fluid limit of the gel
  3. Fluid limit of the forest
  4. Total gelation time and vicinity
  5. Organization of the paper
  6. Frozen Erdős-Rényi and uniform random forests
  7. Background on uniform random forests
  8. Asymptotics of expected conditional jumps
  9. Estimates on the jumps of $G_{p,n}$
  10. Properties of the fluid limit functions
  11. The functions $g_p,d_p,v_p,e_p$ and $r_p$
  12. Approximation
  13. The functions $t_{p,k}$
  14. Convergence to the fluid limit
  15. Preliminaries
  16. ...and 28 more sections

Key Result

Theorem 1.2

As $n \rightarrow \infty$, for the topology of uniform convergence on compacts,

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (76)

  • Definition 1.1
  • Theorem 1.2
  • Proposition 1.3: Free forest property, ContatCurien23,viau25
  • Proposition 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Corollary 1.7: Largest trees
  • Corollary 1.8: Typical tree
  • Theorem 1.9
  • Proposition 1.10
  • ...and 66 more