Publications by Clément Hongler

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We study the 2-dimensional Ising model at critical temperature on a simply connected subset of the square grid Z2. The scaling limit of the critical Ising model is conjectured to be described by Conformal Field Theory; in particular, there is expected to b ...

At initialization, artificial neural networks (ANNs) are equivalent to Gaussian processes in the infinite-width limit [12, 9], thus connecting them to kernel methods. We prove that the evolution of an ANN during training can also be described by a kernel: ...

Critical statistical mechanics and Conformal Field Theory (CFT) are conjecturally connected since the seminal work of Beliavin et al. (Nucl Phys B 241(2):333-380, 1984). Both exhibit exactly solvable structures in two dimensions. A long-standing question ( ...

Random Feature (RF) models are used as efficient parametric approximations of kernel methods. We investigate, by means of random matrix theory, the connection between Gaussian RF models and Kernel Ridge Regression (KRR). For a Gaussian RF model with P feat ...

In this paper, we consider the set of interfaces between + and - spins arising for the critical planar Ising model on a domain with + boundary conditions, and show that it converges to nested CLE3. ...

We introduce the SPRIG (Smart Proofs via Recursive Information Gathering) protocol. SPRIG allows agents to propose, question, and defend mathematical proofs in a decentralized fashion. A structure of stakes and bounties aims at producing debates in good fa ...

This book introduces the mathematical ideas connecting Statistical Mechanics and Conformal Field Theory (CFT). Building advanced structures on top of more elementary ones, the authors map out a well-posed road from simple lattice models to CFTs. Structure ...

Supervised deep learning involves the training of neural networks with a large number N of parameters. For large enough N, in the so-called over-parametrized regime, one can essentially fit the training data points. Sparsitybased arguments would suggest th ...

We prove that crossing probabilities for the critical planar Ising model with free boundary conditions are conformally invariant in the scaling limit, a phenomenon first investigated numerically by Langlands, Lewis and Saint-Aubin (J. Stat. Phys. 98 (2000) ...

We study the probabilistic modeling performed by Autoregressive Large Language Models (LLMs) through the angle of time directionality, addressing a question first raised in (Shannon, 1951). For large enough models, we empirically find a time asymmetry in t ...

— We discuss the magnetization Mm in the m-th column of the zig-zag layered 2D Ising model on a half-plane using Kadanoff–Ceva fermions and orthogonal polynomials techniques. Our main result gives an explicit representation of Mm via m × m Hankel determina ...

The Neural Tangent Kernel is a new way to understand the gradient descent in deep neural networks, connecting them with kernel methods. In this talk, I'll introduce this formalism and give a number of results on the Neural Tangent Kernel and explain how th ...

One key challenge in Artificial Life is designing systems that display an emergence of complex behaviors. Many such systems depend on a high-dimensional parameter space, only a small subset of which displays interesting dynamics. Focusing on the case of co ...

We rigorously prove the existence and the conformal invariance of scaling limits of the magnetization and multi-point spin correlations in the critical Ising model on arbitrary simply connected planar domains. This solves a number of conjectures coming fro ...

Clément Hongler

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Ising Model: Local Spin Correlations and Conformal Invariance

Neural Tanget Kernel: Convergence and Generalization in Neural Networks

Conformal Field Theory at the Lattice Level: Discrete Complex Analysis and Virasoro Structure

Implicit Regularization of Random Feature Models

The Scaling Limit Of Critical Ising Interfaces Is Cle3

Smart Proofs via Recursive Information Gathering: Decentralized Refereeing by Smart Contracts

Lattice Models and Conformal Field Theory

Scaling description of generalization with number of parameters in deep learning

Conformal invariance of crossing probabilities for the Ising model with free boundary conditions

Arrows of Time for Large Language Models

MAGNETIZATION IN THE ZIG-ZAG LAYERED ISING MODEL AND ORTHOGONAL POLYNOMIALS

Neural Tangent Kernel: Convergence and Generalization in Neural Networks (Invited Paper)

Looking for Complexity at Phase Boundaries in Continuous Cellular Automata

Conformal Invariance of Spin Correlations in the Planar Ising Model