Minicourses

The Chalker-Coddington network model of the integer quantum Hall transition

Martin Zirnbauer

Lecture 1: INTEGER QUANTUM HALL BASICS.
1. Hall conductance. 2. High-field limit: guiding center drift and
quantum percolation scenario. 3. Topological phase transition from
bulk-boundary correspondence. 4. Experiments and computer simulations of
the plateau transition.

Lecture 2: NETWORK MODEL.
1. Chalker-Coddington network model (formulation). 2. Double-path
expansion of point-contact conductance. 3. Positivity of path measure in
the Kac-Ward basis. 4. Multifractal scaling of critical states.

Lecture 3: METHODS OF ANALYSIS.
1. Wegner-Efetov supersymmetry method. 2. Color-Flavor Transformation.
3. Read's method: supersymmetric vertex model. 4. Cauchy transform of
the network model.

Lecture 4: CONJECTURES.
1. Pruisken-Khmelnitskii scaling picture. 2. Conformal field theory for
the renormalization-group fixed point at criticality: (i) fixed-point
conductance, (ii) multifractal scaling exponents from Gaussian Free
Field, (iii) singular continuous spectral measure

Lecture 5: RESEARCH SEMINAR.
Wess-Zumino-Witten anomaly of the Chalker-Coddington model

Notes: download here

Reinforcement, supersymmetry and isomorphism theorems

In this course we study the so-called Edge-Reinforced Random Walk (ERRW), Vertex-Reinforced Jump Process, their non-reversible generalizations called *-Edge-Reinforced Random Walk (*-ERRW) and *-Vertex-Reinforced Jump Process (*-VRJP), and their link to supersymmetry and isomorphism theorems.

We start with the Edge-Reinforced Random Walk on a discrete locally finite graph, introduced in the seminal work of Coppersmith and Diaconis (1986), which moves to a nearest-neighbor with a probability proportional to the number of crossings of the corresponding edge. This allows us to define the notions of partial exchangeability in the sense of Diaconis and Freedman (1982), de Finetti’s theorem, and the link with Bayesian statistics.

Then we introduce the *-Edge-Reinforced Random Walk (*-ERRW) through its motivation from Bayesian statistics of variable order Markov Chains. The process is again partially exchangeable, so that it is also a random walk in random environment, which enables one to do similar Bayesian statistics.

Then we define the Vertex-Reinforced Jump Process (VRJP), a continuous time process on a discrete locally finite graph, which moves to an adjacent vertex at a rate proportional to its current local time. We show how this process is related to the ERRW.

We discuss how the mixing measure of the VRJP is related to the so-called supersymmetric hyperbolic sigma field model first studied by Disertori, Spencer and Zirnbauer (2010). We introduce an alternative approach for the study of that measure, through a random Schrödinger approach.

Similarly, the *-ERRW can be associated to a continuous process called the *-Vertex Reinforced Jump Process, which itself is in general not exchangeable, but can be made exchangeable through averaging of the initial local time. We compute the limiting measure.

Finally, we discuss Dynkin and Ray-Knight isomorphism theorems for the VRJP, see Bauerschmidt, Helmuth and Swan (2019, 2021). We introduce a new supersymmetric space, generalizing H^2|2 , called H^5|4*, and we show how it is related to the generalizations of those isomorphisms to the *-VRJP.

Notes: download here

Christophe Sabot & Pierre Tarres

Research Talks

2D Brownian loop soups, conformally invariant fields and height gap

In my talk I will present the joint work with Antoine Jego and Wei Qian, also related to earlier works with Aru and Sepulveda, as well as with Aidekon, Berestycki, Jego. The Brownian loop soups are natural Poisson collections of Brownian loops in a 2D domain, that satisfy a conformal invariance property. In particular, there is a scale invariance property with tiny loops at all small scales. The question of clusters formed by these Brownian loops hase been first studied by Sheffield and Werner. They showed a phase transition at the intensity parameter c=1. For c<= 1, the outer boundaries have been identified as Conformal Loop Ensembles CLE_k. Later, in my joint work with Aru and Sepulveda, the whole clusters at c=1 have been identified as sign components of the 2D Gaussian free field. In the work with Jego and Qian, we focus on the subcritical regime c<1. We show that the one-arm probabilities for clusters behave like | log epsilon|^{-(1-c/2)}. We further show that out of the Minkowski contents of these clusters plus some independent signs, one can construct conformally invariant random fields, which are correlated as a power of log. We prove that all these fields present a height gap phenomenon, similarly to the 2D GFF. We also formulate conjectures on the renormalized powers of these fields.

Titus Lupu

The vertex-reinforced jump processes and a non-linear supersymmetric hyperbolic sigma model with long-range interactions

The vertex-reinforced jump process on the d-dimensional lattice with long-range jumps is transient in any dimension d as long as the initial weights do not decay too fast. The main ingredients in the proof are: an analysis of the corresponding random environment on finite boxes, a comparison with a hierarchical model, and the reduction of the hierarchical model to a non-homogeneous effective one-dimensional model. For the corresponding nonlinear supersymmetric hyperbolic sigma model, a certain marginal of the random field has asymptotically arbitrarily small fluctuations for large enough interactions. Joint work with Margherita Disertori and Franz Merkl.

Silke Rolles

The Monomer Double-Dimer Model through complex spins

We consider a generalization of the double-dimer model that encompasses several models of interest, including the monomer double-dimer model, the dimer model, the Spin O(N) model, and models related to the loop O(N) model. We prove that, on two-dimension-like graphs (such as slabs), both the correlation function and the probability that a loop visits two given vertices decay to zero as the distance between the vertices increases. Our analysis introduces a new (complex) spin representation that applies to all models in this class, and we provide a new proof of the Mermin-Wagner theorem that does not rely on the positivity of the Gibbs measure. Even for the well-studied dimer and double-dimer models, our results are new: since they do not rely on solvability or Kasteleyn’s theorem, they apply beyond the framework of planar graphs. Based on a joint work with Wei Wu.

Lorenzo Taggi

Restrictions of vertex-reinforced jump processes to subgraphs

The restriction of the vertex-reinforced jump process to a subset of the vertex set is a mixture of vertex-reinforced jump processes. A similar statement holds for the corresponding non-linear hyperbolic supersymmetric sigma model. This can be applied to vertex-reinforced jump processes on subdivided versions of graphs of bounded degree, where every edge is replaced by a finite sequence of edges. It yields that discrete-time processes associated to suitable corresponding restrictions are mixtures of positive recurrent Markov chains. Joint work with Margherita Disertori and Silke Rolles.

Notes: download here

Franz Merkl

The variational principle for the interacting Bose gas

The Feynman representation of the interacting Bose gas is conjectured to involve "interacting interlacements" at sufficiently high densities in the thermodynamic limit, making it a challenging ensemble to study. In this talk, I will introduce the Feynman loop representation and explain how we can use it to derive a (probabilistic) variational principle for the pressure of the interacting gas at all densities. The main challenge is to piece together short snippets of paths into longer segments, using large deviation tools and point process theory. This is joint work (in progress) with G. Bellot (University of Lille).

Quirin Vogel

Phase transitions of H^2|2-model on Dyson hierarchical lattice and on Z with decaying potential

Work by Disertori, Merkl, and Rolles has established that the H^2|2-model exhibits long-range order on the Dyson hierarchical lattice for spectral dimensions greater than 2. We show how to establish local order when the spectral dimension is less than 2. In the second part, we discuss the phase transition of the H^2|2-model on Z with a decaying potential. Finally, we survey some open questions concerning the Anderson transition for an operator related to the H^2|2-model.

Xiaolin Zeng

Yves Le Jan

After introducing the relations between spanning trees, random loops and bosonic/fermionic fields on general graphs, we focus on the case of the complete graph and derive a few asymptotics as its size increases to infinity.

Loops, trees and fields on complete graphs

Schwarzian Field Theory for Probabilists

What does Liouville field theory, the SYK random matrix model and JT quantum gravity have in common? If you’d ask a physicist in recent years, they would be quick to point out that the low-energy behaviour of all these models should be described by the Schwarzian field theory. In itself, the latter can be understood as a probability measure on a quotient of the group of circle-diffeomorphisms Diff(T)/PSL(2,R). We discuss a rigorous approach to construct the measure in terms of a non-linear transformation of Brownian bridges, following ideas by Belokurov—Shavgulidze. Furthermore, we present new results that uniquely characterise the measure in terms of an appropriate change-of-variables formula, which can be seen as an analogue of the Cameron—Martin theorem on the space of circle diffeomorphisms. As a byproduct, we also obtain a short proof for the calculation of the measure’s partition function (i.e. total mass), confirming a prediction by Stanford—Witten. This talk is based on joint work with Roland Bauerschmidt and Ilya Losev.

Peter Wildemann

Random forests and the renormalization group

In this talk, I will discuss ongoing work regarding the statistical mechanics of unrooted spanning forests, known also as the arboreal gas. The model has a parameter, beta, controlling the density of edges of a typical sample. With R. Bauerschmidt and T. Helmuth we showed that in dimension at least three there is a percolative phase transition in beta. Moreover, for beta large enough, the model exhibits massless decay of truncated correlations. In dimension two, the story is different. With Bauerschmidt, Helmuth and A. Swan we showed there is no percolative phase transition (modulo boundary condition issues). However, at least conjecturally our result in two dimensions is non optimal: there is expected to be exponential decay of correlations for any value of beta. This conjecture is a special case of the mass gap conjecture for O(n) models, for $n\geq 3.$ With this conjecture in mind, with Bauerschmidt and Helmuth I have been studying the behavior of the arboreal gas in the continuum limit on a two dimensional torus, i.e. with an infrared cutoff. As the mesh of the discrete torus goes to 0, we find the asympototic behavior of \beta so that the density of the component of the origin converges and study various indicators of non-Gaussian behavior for the limiting field theory. In this talk I plan to discuss these results and hopefully how they connect to the mass gap conjecture.

Nick Crawford

Random walk renormalization and weakly conformal symmetries in complex networks

The renormalization group (RG) is a fundamental tool in statistical mechanics that allows to study how the properties of a certain physical system change with its observation scale. Extending it from Lattices to Complex networks is an open challenge. In this talk, we will tackle the problem of random walk renormalization over general graphs, exploiting the notion of discrete harmonic morphisms, and giving a characterization of them in terms of harmonic measures on graphs. Our main result is showing that harmonic morphisms ”send” harmonic measures into one-step probabilities. Moreover, we will elaborate on some interesting connections with Schramm-Loewener Evolution and Conformal Geometry. Finally, we will establish some new metrics, based on harmonic morphisms, that characterize complex networks.

Francesco Maria Guadagnuolo

Crossing walks ensembles, height functions and wetting in lattice spin models

I will review some heuristics links between phenomenon such as wetting in 2D Potts models, delocalization of low temperature tilted interfaces in 3D Ising model, or delocalization of height functions with a slope, and fluctuation of crossing walks ensembles in 2D. Then I will say a few words about rigorous links between the different models, state some rigorous results obtained using those links, and conclude with some (many) open questions.

Sebastien Ott