*Mathew Penrose*

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198506263
- eISBN:
- 9780191707858
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198506263.003.0009
- Subject:
- Mathematics, Probability / Statistics

This chapter contains some known results on connectivity which are used later on. The notion of unicoherence of a simply-connected set is explained and extended to lattices. Peierls (counting) ...
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This chapter contains some known results on connectivity which are used later on. The notion of unicoherence of a simply-connected set is explained and extended to lattices. Peierls (counting) arguments are described for estimating the number of connected sets in the lattice, and elements of (lattice) percolation theory are described. A multiparameter ergodic theorem is given, and the basic theory of continuum percolation is described. Some of the theory of Poisson point processes are recalled, including the superposition, thinning, and scaling theorems.Less

This chapter contains some known results on connectivity which are used later on. The notion of unicoherence of a simply-connected set is explained and extended to lattices. Peierls (counting) arguments are described for estimating the number of connected sets in the lattice, and elements of (lattice) percolation theory are described. A multiparameter ergodic theorem is given, and the basic theory of continuum percolation is described. Some of the theory of Poisson point processes are recalled, including the superposition, thinning, and scaling theorems.

*Hidetoshi Nishimori and Gerardo Ortiz*

- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780199577224
- eISBN:
- 9780191722943
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199577224.003.0007
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

As the spatial dimensionality $d$ decreases, fluctuations become larger and the stability of the low-temperature ordered state deteriorates. The dimensionality where long-range order disappears is ...
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As the spatial dimensionality $d$ decreases, fluctuations become larger and the stability of the low-temperature ordered state deteriorates. The dimensionality where long-range order disappears is known as lower critical dimension. For instance, the Ising model in one dimension does not display long-range order at finite temperatures, however in two dimensions Peierls argument explains why the same model has an ordered phase below a certain critical temperature. If the basic variables and symmetries are continuous as in the $XY$ and Heisenberg models, the (long-range) ordered state at any finite temperature disappears already in two dimensions. This is the result of Mermin-Wagner's theorem. The $XY$ model nevertheless undergoes an unusual phase transition without an onset of long-range order in two dimensions, which is known as the Kosterlitz-Thouless transition. Gauge or local symmetries cannot spontaneously be broken as elucidated by Elitzur's theorem when applied to lattice gauge theories.Less

As the spatial dimensionality $d$ decreases, fluctuations become larger and the stability of the low-temperature ordered state deteriorates. The dimensionality where long-range order disappears is known as lower critical dimension. For instance, the Ising model in one dimension does not display long-range order at finite temperatures, however in two dimensions Peierls argument explains why the same model has an ordered phase below a certain critical temperature. If the basic variables and symmetries are continuous as in the $XY$ and Heisenberg models, the (long-range) ordered state at any finite temperature disappears already in two dimensions. This is the result of Mermin-Wagner's theorem. The $XY$ model nevertheless undergoes an unusual phase transition without an onset of long-range order in two dimensions, which is known as the Kosterlitz-Thouless transition. Gauge or local symmetries cannot spontaneously be broken as elucidated by Elitzur's theorem when applied to lattice gauge theories.