**Málstofa í stærðfræði**

**Fyrirlesari: Sigurður Örn Stefánsson, Háskóla Íslands**

**Titill: Random maps with large faces**

Staðsetning: Tg-227 í Tæknigarði.

Tímasetning: Fimmtudaginn 31. mars 2022, kl. 10:30.

#### Ágrip:

There has been an immense progress in the understanding of random planar maps in the last two decades. An important breakthrough was the independent proofs of Le Gall and Miermont that certain classes of these maps (uniform triangulations and uniform 2p-angulations) converge towards the so called Brownian map. Subsequently there have been many extensions showing that the Brownian map arises as a universal limit of a large family of discrete models. Another important family of random maps are the so called stable maps which arise as limits of random planar maps which are defined in such a way that large faces form in the maps. The study of stable maps is motivated by the conjecture (and in some cases proven fact) that they appear as natural objects when the Brownian map is decorated with statistical mechanical models. To date much less is known about the stable maps than the Brownian map, although there are some exciting results on the horizon.

The focus of the current talk is a model of causal planar maps which was introduced in its original form by Ambjorn and Loll. The limit of the causal maps in the uniform case (which is analogous to the Brownian map case above) turns out to be trivial. However when the measure is tweaked so that large faces are forced to appear, we show that there arises an interesting scaling limit which we call the stable shredded sphere. I will define the stable shredded sphere, describe some of its properties and explain briefly the key ingredients in the proof of the limit result.

This is joint work with Jakob Björnberg and Nicolas Curien. See https://arxiv.org/abs/1912.01378 for details.

**Málstofa í stærðfræði**

**Fyrirlesari: Adam Timar, Háskóla Íslands og Alfréd Rényi stæðfræðistofuna í Budapest **

**Titill: Perfect matchings of optimal tail for random point sets**

Staðsetning: Tg-227 í Tæknigarði.

Tímasetning: Fimmtudaginn 10. mars 2022, kl. 10:30.

#### Ágrip:

Consider two infinite random discrete sets of points in the Euclidean space whose distributions are invariant under isometries. Find a perfect matching between them that makes the distance between pairs decay as fast as possible (in the proper sense). Our setup will be when the random point sets are given by Poisson point processes, and we are interested in *factor* matching rules, meaning that every point can determine its pair using local information and using the same method. In the talk we will introduce all the necessary notions and present the recent solution to the above problem. We will see how the solution is connected to a land-division problem and to the question of whether it is possible to cut a disc of unit area into finitely many pieces and reassemble a unit square from these pieces.

Málstofa í stærðfræði

### Fyrirlesari: Thomas Selig, University of Strathclyde

### Titill: EW-tableaux, permutations and recurrent configurations of the sandpile model on Ferrers graphs.

Staðsetning: VRII, V-147

Tími: Miðvikudagur 27. júní kl. 10:30

### Ágrip:

The Abelian sandpile model (ASM) is a dynamic process on a graph. More specifically, it is a Markov chain on the set of configurations on that graph. Of particular interest are the recurrent configurations, i.e. those that appear infinitely often in the long-time running of the model. We study the ASM on Ferrers graphs, a class of bipartite graphs in one-to-one correspondence with Ferrers diagrams. We show that minimal recurrent configurations are in one-to-one correspondence with a set of certain 0/1 fillings of the Ferrers diagrams introduced by Ehrenborg and van Willigensburg. We refer to these fillings as EW-tableaux, and establish a bijection between the set of EW-tableaux of a given Ferrers diagram and a set of permutations whose descent bottoms are given by the shape of the Ferrers diagram. This induces a bijection between these permutations and minimal recurrent configurations of the ASM. We enrich this bijection to encode all recurrent configurations, via a decoration of the corresponding permutation. We also show that the set of recurrent configurations over all Ferrers graphs of a given size are in bijection with the set of alternating trees of that size.

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### Fyrirlesari: Delphin Sénizergues, Université Paris 13

### Titill: Random metric spaces constructed using a gluing procedure

Staðsetning: VRII-V147

Tími: Mánudagur 18. júní kl. 10:50

### Ágrip:

I will introduce a model of random trees which are constructed by iteratively gluing an infinite number of segments of given length onto each other. This model can be generalized to a gluing of „blocks“ that are more complex than segments. We are interested in the metric properties of the limiting metric space, mainly its Hausdorff dimension. We will show that its Hausdorff dimension depends in a non-trivial (and surprising !) manner on the different scaling parameters of the model and the dimension of the blocks.

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### Fyrirlesari: Adam Timar, Renyi Institute, Budapest

Titill: Allocation rules for the Poisson point process

Staðsetning: Árnagarður 101.

Tími: Föstudagur 3. júní, klukkan 13:20-14:20.

### Ágrip:

Consider the Poisson point process in Euclidean space. We are interested in functions on this random point set whose value in each configuration point is given by some „local“ rule (no „central planning“). One example is the so-called allocation problem, where we want to partition R^d to sets of measure 1 and match them with the point process, in a translation equivariant way. We want to make the allocated set optimal in some sense (e.g., the distribution of the diameter shows fast decay). We will present some allocation schemes, among them one with an optimal tail, which is joint work with R. Marko.

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### Fyrirlesari: Jakob Björnberg

Titill: Random permutations and quantum Heisenberg models

Staðsetning: V-157, VRII.

Tími: Föstudagur 1. apríl kl. 13:20.

### Ágrip:

The interchange process (or random-transposition random walk) is a model for random permutations which is closely related to a model from quantum statistical physics (the ferromagnetic Heisenberg model). In fact, certain ‘cycle-weighted’ interchange processes are equivalent to the latter, and in this talk we present results on such processes. Magnetic ordering in the physical model translates to the occurrence of large cycles in the random permutation.

We focus on the case when the underlying graph is the complete graph (i.e. the ‘mean-field’ case in physical jargon). By a combination of probabilistic techniques and some group character theory we can obtain nice formulas for expectation values in the model, and then use these to identify the critical point.

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### Fyrirlesari: François David

Titill: Planar maps, circle patterns and 2D gravity

Staðsetning: V-157, VRII.

Tími: Föstudagur 11. mars kl. 13:20.

### Ágrip:

I present a model of random planar triangulations (planar maps) based on circle patterns and discuss its properties. It exemplifies the relations between discrete random geometries in the plane, conformally invariant point processes and two dimensional quantum gravity (Liouville theory and topological gravity).

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### Fyrirlesari: Guenter Last

Titill: Second order properties of the Boolean model and the Gilbert graph

Staðsetning: V-157, VRII.

Tími: Föstudagur 30. október, klukkan 15:00-16:00.

### Ágrip:

The Boolean model is a fundamental model of stochastic geometry and continuum percolation. It is a random subset of Euclidean space that arises as the union of random convex grains, independently centered around the points of a stationary Poisson process. The restriction of the Boolean model to a convex and compact observation is a finite union of convex sets. Therefore it makes sense to talk about its intrinsic volumes as volume, surface content, and Euler characteristic.

In this lecture we shall first discuss classical formulae for the densities (normalized expectations) of these intrinsic volumes. Then we proceed with studying asymptotic covariances for growing observation window. These covariances can be expressed in terms of curvature measures associated with a typical grain. In the two-dimensional isotropic case the formulae become surprisingly explicit. We also present a multivariate central limit theorem including Berry-Esseen bounds, derived with the so-called Stein-Malliavin method. If time permits we will also discuss some cluster properties of the Gilbert graph which is a close relative of the Boolean model.

Large parts of the talk are based on joint work with Daniel Hug and

Matthias Schulte.

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### Fyrirlesari: Páll Melsted

Titill: Space Utilization of Cuckoo Hashtables

Staðsetning: V-157, VRII.

Tími: Föstudagur 23. október, klukkan 15:00-16:00.

### Ágrip:

We study the space requirements for Cuckoo Hashing. This can be reduced to

the following question in Random Graphs.

We are given two disjoint sets L,R with |L|=n=alpha*m and |R|=m. We construct a random graph G by allowing each x in L to choose d random neighbours in R. The problem is to find m(G), the size of the largest matching in G.

From the point of view of Cuckoo Hashing, a key question is to locate the threshold for when m(G)=n with high probability, since if m(G) < n it is impossible to store all items. We answer this question exactly for all values of d.

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### Speaker: Hermann Thorisson, University of Iceland

Title: Mass-Stationarity, Shift-Coupling, and Brownian Motion

Staðsetning: Naustið, Endurmenntun (hér)

Tími: Fimmtudagur 28. maí, klukkan 15:00-16:00.

### Ágrip:

After considering mass-stationarity and shift-coupling briefly in an abstract setting, we focus on the special case of stochastic processes on the line associated with diffuse random measures. The main examples are Brownian motion and the Brownian bridge.