## Electronic Journal of Probability

### Scaling limit of ballistic self-avoiding walk interacting with spatial random permutations

#### Abstract

We consider nearest neighbour spatial random permutations on $\mathbb Z ^{d}$. In this case, the energy of the system is proportional to the sum of all cycle lengths, and the system can be interpreted as an ensemble of edge-weighted, mutually self-avoiding loops. The constant of proportionality, $\alpha$, is the order parameter of the model. Our first result is that in a parameter regime of edge weights where it is known that a single self-avoiding loop is weakly space filling, long cycles of spatial random permutations are still exponentially unlikely. For our second result, we embed a self-avoiding walk into a background of spatial random permutations, and condition it to cover a macroscopic distance. For large values of $\alpha$ (where long cycles are very unlikely) we show that this walk collapses to a straight line in the scaling limit, and give bounds on the fluctuations that are almost sufficient for diffusive scaling. For proving our results, we develop the concepts of spatial strong Markov property and iterative sampling for spatial random permutations, which may be of independent interest. Among other things, we use them to show exponential decay of correlations for large values of $\alpha$ in great generality.

#### Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 74, 37 pp.

Dates
Accepted: 27 May 2019
First available in Project Euclid: 3 July 2019

https://projecteuclid.org/euclid.ejp/1562119474

Digital Object Identifier
doi:10.1214/19-EJP328

#### Citation

Betz, Volker; Taggi, Lorenzo. Scaling limit of ballistic self-avoiding walk interacting with spatial random permutations. Electron. J. Probab. 24 (2019), paper no. 74, 37 pp. doi:10.1214/19-EJP328. https://projecteuclid.org/euclid.ejp/1562119474

#### References

• [1] N. Berestycki: Emergence of giant cycles and slowdown transition in random transpositions and k-cycles. Electron. J. Probab. 16, 152-173 (2011).
• [2] V. Betz: Random permutations of a regular lattice. J. Stat. Phys., Volume 155, Issue 6, pp 1222-1248, (2014).
• [3] V. Betz, D. Ueltschi, Y. Velenik: Random permutations with cycle weights. Ann. Appl. Probab. Volume 21, pp 312-331, (2011).
• [4] V. Betz, D. Ueltschi: Spatial random permutations and Poisson-Dirichlet law of cycle lengths. Electron. J. Probab. Volume 16, Issue 41, (2011).
• [5] V. Betz, D. Ueltschi: Spatial random permutations and infinite cycles. Comm. Math. Phys. 285, 469-501 (2009)
• [6] V. Betz, D. Ueltschi: Spatial random permutations with small cycle weights. Probab. Th. Rel. Fields 149, 191-222 (2011).
• [7] V. Betz, D. Ueltschi: Critical temperature of dilute Bose gases. Phys. Rev. A 81, 023611 (2010).
• [8] L. V. Bogachev, D. Zeindler: Asymptotic statistics of cycles in surrogate-spatial permutations. Comm. Math. Phys. 334, 1, p. 39-116.
• [9] M. Campanino, D. Ioffe, and Y. Velenik: Fluctuation theory of connectivities for subcritical random cluster models Ann. Probab., Vol. 36, No. 4 (2008), 1287–1321
• [10] M. Campanino, D. Ioffe, and Y. Velenik: Ornstein-Zernike theory for finite range Ising models above $T_{c}$. Probability Theory and Related Fields 125(3):305-349 (2003)
• [11] M. Campanino and D. Ioffe: Ornstein-Zernike theory for the Bernoulli bond percolation on $\mathbb{Z} ^{d}$. Ann. Probab., Volume 30, Number 2 (2002), 652-682.
• [12] J. T. Chayes and L. Chayes: Ornstein-Zernike Behavior for Self-Avoiding Walks at All Noncritical Temperatures. Comm. Math. Phys. 105, 2, 221-238 (1986)-
• [13] D. Elboim, R. Peled: Limit distributions for Euclidean random permutations. arXiv:1712.03809.
• [14] N. M. Ercolani, D. Ueltschi: Cycle structure of random permutations with cycle weights. Random Struct. Algor. 44, 109-133.
• [15] H. Duminil-Copin, G. Kozma, and A. Yadin: Supercritical self-avoiding walks are space-filling. Ann. Inst. H. Poincaré Probab. Statist., Volume 50, Number 2, pp 315-326, 2014.
• [16] H. Duminil-Copin, A. Glazman, R. Peled,Y. Spinka: Macroscopic loops in the loop $O(n)$ model at Nienhuis’ critical point. ArXiv:1707.09335
• [17] H. Duminil-Copin, R. Peled, W. Samotij, and Y. Spinka: Exponential decay of loop lengths in the loop $O(n)$ model with large $n$. ArXiv: 1412.8326v3. Accepted on Comm. Math. Phys.
• [18] E. Domany, D. Mukamel, B. Nienhuis, and A. Schwimmer: Duality relations and equivalences for models with $O(n)$ and cubic symmetry, Nuclear Physics B 190 (1981), no. 2, 279-287.
• [19] M. Draief and L. Massouli: Epidemics and Rumours in Complex Networks. London Mathematical Society Lecture Note Series: 369.
• [20] R. G. Edwards and A. D. Sokal: Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm. Phys. Rev. D (3) 38 (1988), no. 6, 2009–2012
• [21] D. Gandolfo, J. Ruiz, D. Ueltschi: On a model of random cycles. J. Statist. Phys. 129, 663-676 (2007).
• [22] A. Glazman, I. Manolescu: Uniform Lipschitz functions on the triangular lattice have logarithmic variations. ArXiv: 1810.05592.
• [23] L. Greenberg, D. Ioffe: On an invariance principle for phase separation lines, in Ann. Inst. H. Poincaré Probab. Statist., 41 (2005), no. 5, 871-885.
• [24] S. Grosskinsky, A. A. Lovisolo, D. Ueltschi: Lattice permutations and Poisson-Dirichlet distribution of cycle lengths. J. Statist. Phys. 146, 1105-1121 (2012).
• [25] J. M. Hammersley and D. C. Morton: Poor man’s Monte Carlo. J. Roy. Stat. Soc. B, 16:23-38 (1954).
• [26] J. M. Hammersley: The number of polygons on a lattice. Mathematical Proceedings of the Cambridge Philosophical Society, Volume 57, Issue 3, pp. 516-523 (1961).
• [27] T. E. Harris: Nearest-neighbor Markov interaction processes on multidimensional lattices. Adv. Math. 9, pp. 66-89 (1972)
• [28] D. Ioffe, S. Schlosman, F. Toninelli: Interacting versus entropic repulsion for low temperature Ising polymers, J. Stat. Phys. 158 (2015), no. 5, 1007-1050.
• [29] D. Ioffe: Multidimensional random polymers: a renewal approach, in Random Walks, random fields, and disordered systems, 147-210, Lecture Notes in Math. 2144, Springer, Cham, 2015.
• [30] D. Ioffe: Ornstein-Zernike Behaviour and Analyticity of Shapes for Self-Avoiding Walks On Zd. Markov Processes and Related Fields 4 (1998), 324-350.
• [31] R. Kotecky, P. Milos, D. Ueltschi: The random interchange process on the hypercube. Electron. Comm. Probab. 21, no. 4 (2016).
• [32] Y. Kovchegov: Brownian bridge in percolation, self- avoiding walks and related processes. Ph.D thesis, Department of Mathematics, Stanford University, 2002.
• [33] B. Nienhuis: Exact Critical Point and Critical Exponents of $O(n)$ Models in Two Dimensions. Phys. Rev. Lett., Volume 49, Number 15, (1982).
• [34] N. Madras, G. Slade: The Self-Avoiding Walk. Birkhäuse (2013), reprint of the 1996 Edition. DOI 10.1007/978-1-4614-6025-1.
• [35] L. S. Ornstein, F. Zernike: Proc. Acad. Sci. (Amst.) 17, 793–806 (1915).
• [36] O. Penrose, L. Onsager: Bose-Einstein condensation and liquid Helium. Phys. Rev. 104, 576 (1956)
• [37] O. Schramm: Compositions of random transpositions. Isr. J. Math. 147, 221-243 (2005).
• [38] G. Slade: The self-avoiding walk: a brief survey. in Surveys in Stochastic Processes, J. Blath, P. Imkeller, S. Roelly (eds.), European Mathematical Society (2011).
• [39] L. Taggi: Shifted critical threshold for the loop $O(n)$ model at arbitrarily small $n$, Electronic Comm. Probab. 2018, Vol. 23, paper no. 96, 1-9.
• [40] B. Toth: Improved lower bound on the thermodynamic pressure of the spin $1/2$ Heisenberg ferromagnet. Lett. Math. Phys. 28, 75-84 (1993)
• [41] D. Ueltschi: Relation between Feynman Cycles and Off-Diagonal Long-Range Order. Phys. Rev. Lett. 97, 170601 (2006).