# Self-Avoiding Walks on the UIPQ

• Alessandra Caraceni
• Nicolas Curien
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 300)

## Abstract

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.

## Keywords

Uniform Infinite Planar Quadrangulation Random planar maps Self-avoiding walk Peeling process

## Notes

### Acknowledgments

We thank Jérémie Bouttier for fruitful discussion as well as for providing us with an alternative derivation of (1) based on [8]. We are also grateful to Jason Miller for a discussion about [18, 19] and Sect. 5. Figure 1 has been done via Timothy Budd’s software.

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