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Fluctuations in the random-link matching problem.

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Summary
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We analyzed fluctuations in the random-link matching problem using replica and cavity methods. Our study provides exact expressions for the variance and large deviation function of the optimal cost.

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Area of Science:

  • Statistical physics
  • Combinatorial optimization
  • Random graph theory

Background:

  • The random-link matching problem is a fundamental problem in combinatorial optimization.
  • Understanding the fluctuations of the optimal cost is crucial for analyzing the problem's behavior.
  • Previous studies have explored various aspects of this problem, but a complete understanding of its fluctuations remains elusive.

Purpose of the Study:

  • To investigate the fluctuations of the optimal cost in the random-link matching problem.
  • To derive exact expressions for the variance and the large deviation function of the optimal cost.
  • To compare the effectiveness of the replica and cavity methods in analyzing this problem.

Main Methods:

  • The study employs the replica approach, a powerful technique in statistical physics for analyzing disordered systems.
  • The cavity method, another established technique, is also utilized for its complementary strengths.
  • Both methods are applied to derive analytical expressions for the desired quantities.

Main Results:

  • The exact expression for the variance of the optimal cost is derived using replica arguments.
  • The large deviation function of the optimal cost is derived through two independent methods: the replica method and the cavity method.
  • The consistency of results obtained from both methods validates their application.

Conclusions:

  • The replica and cavity methods provide accurate and consistent results for analyzing the random-link matching problem.
  • The derived expressions for variance and the large deviation function offer deeper insights into the problem's statistical properties.
  • This work contributes to a more comprehensive understanding of random matching problems in statistical physics and optimization.