# One-dimensional long-range percolation: A numerical study.

@article{Gori2017OnedimensionalLP, title={One-dimensional long-range percolation: A numerical study.}, author={Giacomo Gori and Marcus Michelangeli and Nicol{\`o} Defenu and Andrea Trombettoni}, journal={Physical review. E}, year={2017}, volume={96 1-1}, pages={ 012108 } }

In this paper we study bond percolation on a one-dimensional chain with power-law bond probability C/r^{d+σ}, where r is the distance length between distinct sites and d=1. We introduce and test an order-N Monte Carlo algorithm and we determine as a function of σ the critical value C_{c} at which percolation occurs. The critical exponents in the range 0<σ<1 are reported. Our analysis is in agreement, up to a numerical precision ≈10^{-3}, with the mean-field result for the anomalous dimension…

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## References

SHOWING 1-10 OF 44 REFERENCES

The diameter of a long range percolation graph

- MathematicsSODA 2002
- 2002

One can model a social network as a long-range percolation model on a graph {0, 1, …, N}2. The edges (x, y) of this graph are selected with probability ≈ β/||x - ys if ||x - y|| > 1, and with…

MATH

- 1992

Abstract: About a decade ago, biophysicists observed an approximately linear relationship between the combinatorial complexity of knotted DNA and the distance traveled in gel electrophoresis…

Elements of Phase Transitions and Critical Phenomena

- Mathematics
- 2011

1. Phase transitions and critical phenomena 2. Mean-field theories 3. Renormalization group and scaling 4. Implementation of the renormalization group 5. Field theory 6. Conformal field theory 7.…

Statistical Field Theory: An Introduction to Exactly Solved Models in Statistical Physics

- Physics
- 2009

I: INTRODUCTORY NOTIONS II: BIDIMENSIONAL LATTICE MODELS III: QUANTUM FIELD THEORY AND CONFORMAL VARIANCE IV: AWAY FROM CRITICALITY

"J."

- 1890

however (for it was the literal soul of the life of the Redeemer, John xv. io), is the peculiar token of fellowship with the Redeemer. That love to God (what is meant here is not God’s love to men)…

Physics of Long-Range Interacting Systems

- Physics
- 2014

PART I: STATIC AND EQUILIBRIUM PROPERTIES PART II: DYNAMICAL PROPERTIES PART III: APPLICATIONS

Ann

- 2005

Aaron Beck’s cognitive therapy model has been used repeatedly to treat depression and anxiety. The case presented here is a 34-year-old female law student with an adjustment disorder with mixed…

Phys

- Rev. E 92, 052113
- 2015

Phys

- Rev. D 92, 025012
- 2015

A 16

- L639
- 1983