21 June 2018



Speakers: Timothy Budd, Wioletta Ruszel, Tim van de Brug, Federico Camia.

Date:     June 21, 2018

Location: VU University Amsterdam, Science Building, Room WN-S631

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10:15 – 11:15 Timothy Budd (Radboud University Nijmegen)

Title: Lessons from the O(n) loop model on a random surface

Abstract: The O(n) loop models describe probability distributions on collections of non-intersecting loops on the two-dimensional lattice, which are conjectured to possess scaling limits described by conformal loop ensembles. In this talk I will discuss the combinatorics and some statistics of the O(n) loop model when the regular lattice is replaced by a random planar map. I will discuss two types of connections between this work and loop-type objects in the plane. The first relies on the uniformization of the random planar map, which is conjectured to scale towards Liouville quantum gravity together with an independent conformal loop ensemble. The second relates the O(n) loop model on a planar maps to the combinatorics of walks on the square lattice. This connection allows one to determine exact winding angle statistics of uniform random loops
(self-intersecting this time) on the square lattice.

11:30 – 12:30 Wioletta Ruszel (TU Delft)

Title: Sandpile models, random interfaces and scaling limits

Abstract: In this talk we will introduce a special probabilistic cellular automaton, namely the (divisible) sandpile model and discuss connections with random interfaces and their (continuous space) scaling limits.

The divisible sandpile model (DSM) is the continuous height counterpart of the Abelian sandpile which is a toy model introduced by Bak, Tang and Wiesenfeld in 1987 displaying self-organized criticality. Self-organized critical models are in some sense driving themselves into a critical state without fine-tuning of any external parameters such as the temperature.

DSM were used in connections with internal diffusion limited aggregation growth models. In the DSM setting one is starting with a random continuous initial height configuration and if the height exceeds 1, keeping mass 1 and  toppling the excess to some neighbours with equal probability. It turns out that under some conditions the final configuration will be the all 1 configuration and that the odometer which records the amount of mass emitted during the stabilization is defining a random interface configuration. We will discuss this
construction and under which conditions the random interface is of Gaussian, alpha-stable or fractional type and determine their continuous scaling limit.

This is joint work with: Alessandra Cipriani (TU Delft), Rajat Hazra (ISI Kolkata) and Leandro Chiarini (TU


12:30 – 14:00 Lunch Break


14:00 – 15:00 Tim van de Brug (VU Amsterdam)

Title: Random fields in medical image segmentation

Abstract: Multiple sclerosis is a disease in which parts of the brain are damaged, resulting in a disability of the nervous system to communicate. White matter lesions in the brain and spinal cord are the most important biomarkers to monitor disease progression. Manual identification of white matter lesions on MRI scans uses expert knowledge and is therefore not possible at large scale. We study semi-automated image segmentation methods that combine machine learning algorithms with human input from a large number of non-expert raters. In this talk we will highlight some interesting connections between the segmentation algorithms and spatial probabilistic models. This is joint work with Hugo Vrenken and Soheil Damangir (Dept. of Radiology, VUmc).


15:15 – 16:15 Federico Camia (NYU Abu Dhabi and VU Amsterdam)

Title: The 2D Ising Magnetization Field

Abstract: In this talk, I will present recent results on the critical scaling limit of the magnetization field in the two-dimensional Ising model, as well as on the near-critical scaling limit with an external magnetic field. For both the critical and the near-critical case, Camia, Garban and Newman showed that the lattice magnetization, properly normalized, converges to a continuum field, i.e. a random generalized function. In the critical case, the limiting magnetization field is conformally covariant. In the near-critical case, Camia, Jiang and Newman showed exponential decay of the truncated two-point function, proving the existence of a mass gap for the limiting field theory. They also showed that the mass M, i.e. the inverse of the correlation
length, scales with the renormalized external field h like log M= (8/15) log h + c for some constant c.


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