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# Self-Avoiding Walks on the UIPQ

__Publication__

handle: 11384/125564

# Self-Avoiding Walks on the UIPQ

We study an annealed model of Uniform Infinite Planar Quadrangulation (UIPQ) with an infinite two-sided self-avoiding walk (SAW), which can also be described as the result of glueing together two independent uniform infinite quadrangulations of the half-plane (UIHPQs). We prove a lower bound on the displacement of the SAW which, combined with the estimates of [15], shows that the self-avoiding walk is diffusive. As a byproduct this implies that the volume growth exponent of the lattice in question is 4 (as is the case for the standard UIPQ); nevertheless, using our previous work [9] we show its law to be singular with respect to that of the standard UIPQ, that is – in the language of statistical physics – the fact that disorder holds.

- Université Paris Diderot France
- University of Paris-Saclay France
- Scuola Normale Superiore di Pisa Italy
- University of Bath United Kingdom

Microsoft Academic Graph classification: Pure mathematics Upper and lower bounds Combinatorics Planar Volume growth Lattice (order) Exponent Self-avoiding walk Mathematics

ACM Computing Classification System: ComputingMilieux_MISCELLANEOUS

Peeling proce, Random planar map, Settore MAT/06 - Probabilita' e Statistica Matematica, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Uniform Infinite Planar Quadrangulation, Self-avoiding walk, [MATH]Mathematics [math]

Peeling proce, Random planar map, Settore MAT/06 - Probabilita' e Statistica Matematica, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Uniform Infinite Planar Quadrangulation, Self-avoiding walk, [MATH]Mathematics [math]

Microsoft Academic Graph classification: Pure mathematics Upper and lower bounds Combinatorics Planar Volume growth Lattice (order) Exponent Self-avoiding walk Mathematics

ACM Computing Classification System: ComputingMilieux_MISCELLANEOUS

###### 7 Research products, page 1 of 1

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- Université Paris Diderot France
- University of Paris-Saclay France
- Scuola Normale Superiore di Pisa Italy
- University of Bath United Kingdom

We study an annealed model of Uniform Infinite Planar Quadrangulation (UIPQ) with an infinite two-sided self-avoiding walk (SAW), which can also be described as the result of glueing together two independent uniform infinite quadrangulations of the half-plane (UIHPQs). We prove a lower bound on the displacement of the SAW which, combined with the estimates of [15], shows that the self-avoiding walk is diffusive. As a byproduct this implies that the volume growth exponent of the lattice in question is 4 (as is the case for the standard UIPQ); nevertheless, using our previous work [9] we show its law to be singular with respect to that of the standard UIPQ, that is – in the language of statistical physics – the fact that disorder holds.