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Related Experiment Video

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Large-scale Reconstructions and Independent, Unbiased Clustering Based on Morphological Metrics to Classify Neurons in Selective Populations

Large-scale Reconstructions and Independent, Unbiased Clustering Based on Morphological Metrics to Classify Neurons in Selective Populations

Published on: February 15, 2017

Linked cluster expansion on trees.

Deepak Iyer¹, Yuyi Wan²

¹Department of Physics & Astronomy, Bucknell University, 1 Dent Dr, Lewisburg, Pennsylvania 17837, USA.

Physical Review. E

|March 18, 2023

Summary

The linked cluster expansion efficiently calculates properties for lattice models. This study extends the method to Cayley and Bethe lattices, revealing its effectiveness and limitations for these tree-like structures.

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Area of Science:

Statistical Mechanics
Condensed Matter Physics
Computational Physics

Background:

The linked cluster expansion is a powerful technique for analyzing lattice models.
Its efficiency in 1D and 2D systems is well-established.
Application to tree-like lattices like Cayley and Bethe has been unexplored.

Purpose of the Study:

To develop and apply the linked cluster expansion to Cayley and Bethe lattices.
To investigate the convergence efficiency and limitations of the method on these structures.
To gain insights into the behavior of disordered systems and phase transitions.

Main Methods:

Developing a novel linked cluster expansion for treelike lattices.
Mapping finite treelike clusters to one-dimensional finite chains.
Analyzing the impact of lattice constants and boundary conditions.

Main Results:

Demonstrated that finite treelike clusters map to 1D chains for specific nearest-neighbor Hamiltonians.
Identified differing lattice constants as the cause for distinctions between Cayley and Bethe lattices.
Derived closed-form formulas for zero-field susceptibility and partition functions on Bethe lattices.
Showcased the linked cluster expansion's ability to eliminate boundary terms, explaining its convergence efficiency.

Conclusions:

The linked cluster expansion is effectively extended to Cayley and Bethe lattices.
The method's efficiency is attributed to its inherent elimination of boundary effects.
Results provide a foundation for further studies on disordered systems and lattice models.