The SLE/CLE Continuum Perspective on the Two-dimensional Critical Ising Model

• Wendelin WERNER (ETH Zürich)

Room: Marilyn and James Simons Conference Center I will survey recent and less recent aspects of the description of the scaling limit of the two-dimensional critical Ising model in terms of Conformal Loop Ensembles, which are the loop ensemble versions of the Schramm-Loewner Evolutions. In particular, we will try to illustrate the fact that many of the ideas that had emerged in the study of the discrete models appear also directly in this continuum setting that focuses on random geometric curves.

Non-self-averaging in the Critical Point of the 2D Random Ising Model

• Victor DOTSENKO (LPTMC Jussieu)

Room: Marilyn and James Simons Conference Center In this talk, I present a brief review of the recent results on sample-to-sample fluctuations in a critical two-dimensional Ising model with quenched random ferromagnetic couplings. Using replica calculations in the renormalization group framework I derive explicit expressions for the probability distribution function (PDF) for the critical internal energy and for as well as for the specific heat fluctuations. For the singular part of the internal energy, it is shown that in the critical point both its average value and the typical value of its sample-to-sample fluctuations scale with the system size $L$ like $\sim L \ln\ln(L)$ which implies that the internal energy of disordered 2D ferromagnetic Ising model is non-self averaging in the critical point. In contrast to that, the specific heat is shown to be self-averaging with a distribution function that tends to a $\delta$-peak in the thermodynamic limit $L \to \infty$.

The Scaling Limit of Non-solvable 2D Ising Models via Fermionic RG

• Alessandro GIULIANI (Università di Roma 3)
• Rafael GREENBLATT (Universita' degli Studi Roma Tre)

Room: Marilyn and James Simons Conference Center The scaling limit of any 2D Ising model with ferromagnetic short-range interactions at the critical point is expected to be a Conformal Field Theory with c=1/2, one instance of which is the theory of free Majorana fermions. This expectation comes with extremely detailed predictions on critical exponents, on the form of the scaling limit of multipoint correlations, and on their conformal covariance in finite domains. While the conjectured picture is now fully proved for nearest-neighbor interactions, many open problems remain in the case of more general interactions. We will review the history of the problem and the state-of-the-art in the context of 2D Ising models with nearest-neighbor interactions plus weak additional finite range interactions, focusing on results proved by rigorous fermionic Renormalization Group methods, and we will discuss perspectives and open problems. Based on joint works with Vieri Mastropietro.

On Crossing Probabilities in Critical Random-cluster Models

• Eveliina PELTOLA (HCM University of Bonn)

Room: Marilyn and James Simons Conference Center I will discuss exact solvability results (in a sense) for scaling limits of interface crossings in critical random-cluster models in the plane with various boundary conditions. The results are rigorous for the FK-Ising model, Bernoulli percolation, and the spin-Ising model inappropriate setups. The scaling limit formulas describe structures in the corresponding boundary conformal field theory. (Based on joint works with Yu Feng, Mingchang Liu, and Hao Wu - all at Tsinghua University, China).

Random Quantum Ising Spin Chains

• Daniel FISHER (Stanford University)

Room: Marilyn and James Simons Conference Center Random transfer field Ising spin chains are a prototypical example of the interplay between quenched randomness and quantum fluctuations. An approximate real-space renormalization group analysis that becomes exact near the phase zero-temperature phase transition will be presented. Scaling functions and other properties can be computed exactly. Applications of the method to Sinai random walks in random environments, and to higher dimensional random quantum Ising models will be mentioned.

Perspectives on the Renormalisation Group Approach

• Roland BAUERSCHMIDT (University of Cambridge)

Room: Marilyn and James Simons Conference Center The goal of this talk is to review some of the successes but also the outstanding challenges of the renormalisation group approach to the Ising and \varphi^4 models. I will also try to describe a common perspective of the usual approach to the renormalisation group based on perturbation theory and cluster expansions with some of the substitutes for these, based on random currents or random walks.

2D Ising Model and its Tricritical Version, when Theory Meets Experiments

• Giuseppe MUSSARDO (SISSA Trieste)

Room: Marilyn and James Simons Conference Center The magnetic deformation of the 2D Ising model and the thermal deformation of the Tricritical Ising Model is related to the exceptional E_8 and E_7 Lie algebras. The corresponding exact S-matrix theories and the related dynamical structure factors of both models have rich spectroscopy which can be challenged by experiments based on neutron scattering. While in the case of the Ising Model there are nowadays very precise experimental confirmations of the theory, the corresponding experimental set-up of the Tricritical Ising Model is an interesting open question in the field of integrable models.

Gibbs States for (long-range) Ising Models

• Loren COQUILLE (Université Grenoble-Alpes)

Room: Marilyn and James Simons Conference Center I will review old and present new results on standard and long-range Ising models in dimensions 1, 2, and 3. I shall focus on fluctuations or rigidity of interfaces at low temperatures, in the coexistence regime. Based on works in collaboration with Y. Velenik (Geneva) on one hand, A. van Enter (Groningen), A. Le Ny (Paris), and W. Ruszel (Utrecht) on the other hand, ongoing works with R. Durand (Grenoble).

Ising Model, (Para)Fermions, and Field Theory

• Clément HONGLER (EPFL)

Room: Marilyn and James Simons Conference Center In the last 20 years, parafermionic observables have allowed one to rigorously connect lattice models and conformal field theories. I'll present old and recent results and discuss new perspectives (there will new pictures!).

Open pb session Room: Marilyn and James Simons Conference Center

Ising model, Glauber Dynamics and Random Tilings

• Fabio TONINELLI (Technical University of Vienna)

Room: Marilyn and James Simons Conference Center In this talk, I will give a panorama of results for the zero-temperature Glauber dynamics of the 3-dimensional (classical) Ising model. It is well known that, with suitable Dobrushin-type boundary conditions, the Boltzmann-Gibbs distribution of a 3d Ising interface at zero temperature coincides with the uniform measure on rhombus tilings of a certain finite (but large) domain D of the plane. In the same situation, the Glauber dynamics can be seen as a Markov evolution on the set of tilings of D. The holy grail conjecture in this respect, suggested by an "anisotropic mean-curvature flow" heuristics for the interface motion, is that the mixing time of the (continuous-time) dynamics is of order L^{2+o(1)}, with L the diameter of the domain. I will present old and new results that prove this conjecture under the assumption that the asymptotic limit shape in D (that describes the non-random, typical shape of the Ising interface, for L\to\infty) of has no facets. Based on joint works with B. Laslier, as well as on older works with P. Caputo and F. Martinelli

Entropic Repulsion in 3D Ising

• Eyal LUBETZKY (Courant Institute, NYU)

Room: Marilyn and James Simons Conference Center Fifty years ago, Dobrushin famously showed that the 3D Ising interface on a cylinder with plus/minus boundary conditions is rigid. By now there is a detailed understanding of the (2+1)D Solid-On-Solid model that approximates said interface, and notably, its entropic repulsion phenomenon above a hard wall. We will discuss the picture in the SOS approximation and recent progress in confirming these predictions for the 3D Ising model. Based on joint works with Caputo, Martinelli, Toninelli, and Sly on the SOS model, and with Gheissari on the 3D Ising model.

Nonperturbative Analysis of Noncritical Ising Models: Some Applications of the Ornstein–Zernike Theory

• Yvan VELENIK (Université de Genève)

Room: Marilyn and James Simons Conference Center In its modern incarnation (developed during the last two decades), Ornstein-Zernike's theory enables a non-perturbative analysis of non-critical ferromagnetic Ising models (and other models). I'll review some of its recent applications to the asymptotics of correlation functions (in any dimension) and to the fluctuation theory of interfaces (in the planar model).

Using the Ising Model to Explore the Confining Regime of Lattice Gauge Theories

• Michele CASELLE (Università di Torino)

Room: Marilyn and James Simons Conference Center Understanding the physical mechanisms behind confinement is one of the most important open problems in Lattice Gauge Theories (LGTs). In this talk, we discuss two exemplary applications of the Ising model to this problem. In the first example, we study the quark-antiquark correlator in the LGT with SU(2) gauge symmetry. We show that at high temperature, in the neighborhood of the deconfinement transition but still in the confining phase, this correlator can be mapped into the spin correlator of the Ising model and exploit the precise knowledge we have of this correlator to predict the behavior of the confining string. In the second we study the behavior of interfaces in the 3d Ising model which, using duality, can be mapped into the behavior of a (closed) confining string.

Three Dimensional Ising Model as a Non-critical String Theory

• Ara SEDRAKYAN (Yerevan Physics Institute)

Room: Marilyn and James Simons Conference Center I will discuss the sign factor problem in the 3D gauge Ising model, and present the corresponding fermionic model on random surfaces, which leads to the formulation of non-critical fermionic string theory on the basis of induced Dirac action. I will demonstrate how the sign factor model is linked to ordinary and spin quantum Hall plateau transitions, tying them also to non-critical string theory. Based on the sign-factor model new type of matrix model will be formulated, which allows consideration of any spin chain models on random surfaces. This approach opens the way to cross the c=1 barrier in non-critical string theory.

Metric Graph Extensions of Lattice Models with Applications in Stat Mech and Quantum Systems

• Michael AIZENMAN (Princeton University)

Room: Marilyn and James Simons Conference Center As a counterpoint to "be wise and discretize’’, continuous extensions are relevant and provide useful perspective. They occasionally pose challenges but also yield new tools. Examples of both may be found in: the contact process as extension of discrete percolation, long-range 1D Ising and percolation models, the quantum Ising model, quantum spin chains, influence propagation in the random-field Ising model estimated through a stopping time argument, extensions of discrete random height functions, and new results for the Villain O(2) spin system though its metric graph representation.

How the Lizard Got its Colors

• Stanislav SMIRNOV (University of Geneva)

Room: Marilyn and James Simons Conference Center We will discuss how a Turing's reaction-diffusion process in a biological context leads to a rather surprising appearance of Ising-like colorings of the skin of Mediterranean lizards.

Round table: Ruminations on the Ising Model: Past, Present, Future

• Joel LEBOWITZ (Rutgers University)
• Arthur JAFFE (Harvard University)
• Jürg FRÖHLICH (ETH Zürich)
• Tom SPENCER (IAS)
• Geoffrey GRIMMETT (University of Cambridge)

Room: Marilyn and James Simons Conference Center Moderator: Geoffrey GRIMMETT

The Ising Model, the Saga of the Critical Exponents

• Jean ZINN-JUSTIN (CEA Saclay)

Room: Marilyn and James Simons Conference Center The Ising model, being one of the simplest statistical systems, the properties of its phase transition have been studied very early. Of particular interest are its critical exponents. The first guess was obtained from the mean-field approximation. However, starting with Onsager, the exact values in two dimensions could be calculated are were found non-mean-field like. In three dimensions, using various mathematical techniques, approximate values were extracted from the high-temperature expansion. The first calculated values were somewhat biased by not taking into account possible confluent singularities at the critical temperature. The breakthrough came from the renormalization group (RG) method (Wilson). It showed the existence of confluent singularities and led to improved exponent estimates. Moreover, soon, as a solution to the RG equations, and quantum field techniques, exponents could be calculated as epsilon=4-D expansions but also as perturbative expansions n the interaction strength. It took some time to generate a long enough series. Moreover, in both cases, large order estimates showed that the series was always divergent, in the mathematical sense. Original summation methods based on Borel transformation and conformal mapping could be found that led to the first precise exponent estimates.

Emergent Planarity in Two-dimensional Ising Models with Finite-range Interactions

• Vincent TASSION (ETH Zürich)

Room: Marilyn and James Simons Conference Center The boundary spin correlations for planar Ising models have a well-known Pfaffian structure. For Ising models on the square lattice with finite-range interactions, the corresponding graph is not planar and the Pfaffian structure no long holds. Nevertheless, at criticality, the Pfaffian structure of boundary correlations emerges asymptotically (when boundary points are taken far apart). The proven statement establishes an aspect of universality in two dimensions beyond the solvable cases. In this talk, I will present this result and discuss the main ideas of proof, that involve a percolation interpretation of the problem (via Aizenman random currents, and FK percolation) and recent progress in percolation theory (robust Russo-Seymour-Welsh theory). This talk is based on a joint work with Michael Aizenman, Hugo Duminil-Copin and Simone Warzel.

Anatomy of the Ising Model from Conformal Bootstrap

• Alessandro VICHI (Università di Pisa)

Room: Marilyn and James Simons Conference Center The Ising model is the one the simplest and yet non-trivial Conformal Field Theory. For decades it has been a dream to study such an intricate strongly coupled theory non-perturbatively using symmetries and other consistency conditions. This idea, called the conformal bootstrap, saw some successes in two dimensions but it is only in the last ten years that it has been fully realized in three, four, and other dimensions of interest. This renaissance has been possible due to both significant analytical progress in understanding how to set up the bootstrap equations and the development of numerical techniques for finding or constraining their solutions. In this talk we will review the main developments that have led to precise determinations of critical exponents and correlation function coefficients in the Ising model in thee dimensions.

The Multinomial Ising Model

• Rick KENYON (Yale University)

Room: Marilyn and James Simons Conference Center The multinomial Ising model on a graph G=(V,E) is the Ising model on the N-fold “blow-up” G_N of G, whose vertices are V X [N], and edges connect (u,i) to (v,j) if u and v are adjacent. In the limit of large N, we find the critical temperature, phase transitions, conformal invariance properties at criticality, and limit shapes. This is joint work with Cosmin Pohoata.