# Random Structures in High Dimensions (16w5085)

## Organizers

Frank den Hollander (University of Leiden)

David Brydges (University of British Columbia)

## Objectives

The main objective is to communicate the different methods and mathematical traditions that are currently being used and explored to solve the problems. These problems have two levels: (i) interactions are strong (focus on phase diagrams); (ii) interactions are weak (focus on critical exponents).

- Level (i) is well exemplified by polymers chains in random environments, in which there is a competition between maximising the number of polymer configurations and minimising the energy of interaction with the environment. This leads to new problems in the theory of large deviations. A recent example is [4], where the phase transition is characterised by a variational problem that comes from a new type of large deviation problem solved in [3].
- Level (ii) is instanced by the problem of determining the expected end-to-end distance of self-avoiding walk. In [7] a large deviation argument is combined with renewal theory to prove a first result on the end-to-end distance of self-avoiding walk in three dimensions.

For high dimensions the lace expansion, which is an advanced form of the venerable inclusion/exclusion principle, remains a powerful general method. For instance, in dimensions greater than eight, the lace expansion was used in [6,14,15] to show that if random lattice trees are given a time component through the distance to the root, then the resulting rescaled random processes converge to a measure-valued process called super-Brownian motion. Recently, in [16] the lace expansion is generalised to the Ising model and used to determine the asymptotics of the two-point function for the $phi^{4}$ continuous-spin systems and Ising model in dimension larger than four. There is also a remarkable combinatorial application [5] to enumerate three dimensional self-avoiding walks with as many as 30 steps, beating the previous record of 26 steps. The lace expansion has been an analytic tool, but this is instead an application of its algebraic structure. Finally, the construction of the incipient percolation cluster and the elucidation of its properties in [12, 13] is a great achievement in the lace expansion genre.

For self-avoiding walk and spin systems, four dimensions is especially interesting, because it is the borderline where the interactions start to influence the scaling limit. The articles [1], [2] show that self-avoiding-walk and spin systems in four dimensions have critical exponents that differ from mean-field theory by precisely specified logarithmic corrections. These papers import into the probabilistic community rigorous methods coming from quantum-field theory.

Super-Brownian motion mentioned above can be viewed as a singular solution to a nonlinear stochastic PDE. The theory of rough paths [10] is a general theory of singular solutions to nonlinear stochastic PDEs, and this has connections with quantum field theory. A rather central example is the dynamical $phi_{3}^{4}$ problem [11], which is a stochastic PDE whose invariant measure is the $phi_{3}^{4}$ euclidean quantum field theory. The virtue of the rough path theory in this context is that it has very good continuity properties with respect to approximation of stochastic PDEs by more regular PDEs, or even systems of ODEs. Indeed, one can prove that the limiting dynamics obtained from finite-difference approximations do not remember the orientation of the finite-difference lattice. Thus, the invariant measure is proved to be Euclidean invariant, which is one of the axioms of quantum field theory. This axiom hitherto was very difficult to prove starting from a lattice regularisation.

The quantum-field theoretic methods capture some of the theoretical physics understanding of critical phenomena. A very major recent result in this direction is the work of Falco [8], [9], who until his untimely death in 2014 was developing a rigorous understanding of the Coulomb system in two dimensions at its critical point. This is an absolutely fundamental model, because it is believed to contain the scaling limits of many of the exactly solvable models in two dimensions, such as the eight-vertex model.

In summary, the workshop addresses several different ideas and recent progress, and the goal is to assimilate these. It is fortunate that we can at the same time celebrate the birthday of one of the leaders in the field, Gordon Slade.

#### Bibliography

- R. Bauerschmidt, D.C. Brydges, and G. Slade. Logarithmic correction for the susceptibility of the 4-dimensional weakly self-avoiding walk: a renormalisation group analysis. Preprint, (2014).
- R. Bauerschmidt, D.C. Brydges, and G. Slade. Scaling limits and critical behaviour of the $4$-dimensional $n$-component $|varphi|^4$ spin model. Preprint, (2014). To appear in
*J. Stat. Phys.* - Matthias Birkner, Andreas Greven, and Frank den Hollander. Quenched large deviation principle for words in a letter sequence.
*Probab. Theory Related Fields*, 148(3-4):403--456, 2010. - Dimitris Cheliotis and Frank den Hollander. Variational characterization of the critical curve for pinning of random polymers.
*Ann. Probab.*, 41(3B):1767--1805, 2013. - N. Clisby, R. Liang, and G. Slade. Self-avoiding walk enumeration via the lace expansion.
*J. Phys. A: Math. Theor.*,**40**:10973--11017, (2007). - E. Derbez and G. Slade. The scaling limit of lattice trees in high dimensions.
*Commun. Math. Phys.*,**193**:69--104, (1998). - H. Duminil-Copin and A. Hammond. Self-avoiding walk is sub-ballistic.
*Commun. Math. Phys.*,**324**:401--423, (2013). - P. Falco. Kosterlitz--Thouless transition line for the two dimensional Coulomb gas
*Commun. Math. Phys.*,**312**:559--609, (2012). - P. Falco. Critical exponents of the two dimensional coulomb gas at the Berezinskii-Kosterlitz-Thouless transition. http://arxiv.org/abs/1311.2237, 2013.
- Martin Hairer. Introduction to regularity structures. http://arxiv.org/abs/1401.3014.
- Martin Hairer. Singular stochastic PDEs. http://arxiv.org/abs/1403.6353.
- T. Hara and G. Slade. The incipient infinite cluster in high-dimensional percolation.
*Electron. Res. Announc. Amer. Math. Soc.*,**4**:48--55, (1998).

`http:$//$www.ams.org$/$era$/$`. - Markus Heydenreich, Remco van der Hofstad, and Tim Hulshof. High-dimensional incipient infinite clusters revisited.
*J. Stat. Phys.*, 155(5):966--1025, 2014. - M. Holmes. Convergence of lattice trees to super-Brownian motion above the critical dimension.
*Electr. J. Probab.*,**13**:671--755, (2008). - M. Holmes, R. van der Hofstad, and E. Perkins. A criterion for convergence to super-brownian motion on path space. to appear in Ann. Prob., 2014.
- A. Sakai. Application of the lace expansion to the $varphi^4$ model. Preprint, (2014).