Random clusters in the Villain and XY models

In the Ising and Potts model, random cluster representations provide a geometric interpretation to spin correlations. We discuss similar constructions for the Villain and XY models, where spins take values in the circle, as well as extensions to models with different single site spin spaces. In the Villain case, we highlight natural interpretation in terms of the cable system extension of the model. We also list questions and open problems on the cluster representations obtained in this fashion.Comment: 29 pages, 4 figure

Liouville quantum gravity from random matrix dynamics

We establish the first connection between $2d$ Liouville quantum gravity and natural dynamics of random matrices. In particular, we show that if $(U_t)$ is a Brownian motion on the unitary group at equilibrium, then the measures $$|\det(U_t - e^{i \theta})|^{\gamma} dt d\theta$$ converge in the limit of large dimension to the $2d$ LQG measure, a properly normalized exponential of the $2d$ Gaussian free field. Gaussian free field type fluctuations associated with these dynamics were first established by Spohn (1998) and convergence to the LQG measure in $2d$ settings was conjectured since the work of Webb (2014), who proved the convergence of related one dimensional measures by using inputs from Riemann-Hilbert theory. The convergence follows from the first multi-time extension of the result by Widom (1973) on Fisher-Hartwig asymptotics of Toeplitz determinants with real symbols. To prove these, we develop a general surgery argument and combine determinantal point processes estimates with stochastic analysis on Lie group, providing in passing a probabilistic proof of Webb's $1d$ result. We believe the techniques will be more broadly applicable to matrix dynamics out of equilibrium, joint moments of determinants for classes of correlated random matrices, and the characteristic polynomial of non-Hermitian random matrices.Comment: v3: fixes a minor error in the proof of Proposition 3.4, 40 page

Asymptotic analysis of downlink MIMO systems over Rician fading channels

In this work, we focus on the ergodic sum rate in the downlink of a single-cell large-scale multi-user MIMO system in which the base station employs N antennas to communicate with $K$ single-antenna user equipments. A regularized zero-forcing (RZF) scheme is used for precoding under the assumption that each link forms a spatially correlated MIMO Rician fading channel. The analysis is conducted assuming $N$ and $K$ grow large with a non trivial ratio and perfect channel state information is available at the base station. Recent results from random matrix theory and large system analysis are used to compute an asymptotic expression of the signal-to-interference- plus-noise ratio as a function of the system parameters, the spatial correlation matrix and the Rician factor. Numerical results are used to evaluate the performance gap in the finite system regime under different operating conditions.Comment: 5 pages, 2 figures. Published at the 41st IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP 2016), Shanghai, 20-25 March 201

A renormalization approach to the Liouville quantum gravity metric

This thesis explores metric properties of Liouville quantum gravity (LQG), a random geometry with conformal symmetries introduced in the context of string theory by Polyakov in the 80’s. Formally, it corresponds to the Riemannian metric tensor “e^{γh}(dx² + dy²)” where h is a planar Gaussian free field and γ is a parameter in (0, 2). Since h is a random Schwartz distribution with negative regularity, the exponential e^{γh} only makes sense formally and the associated volume form and distance functions are not well-defined. The mathematical language to define the volume form was introduced by Kahane, also in the 80’s. In this thesis, we explore a renormalization approach to make sense of the distance function and we study its basic properties

Weak LQG metrics and Liouville first passage percolation

Funder: Columbia University Minerva fundAbstract: For γ∈(0, 2), we define a weakγ-Liouville quantum gravity (LQG) metric to be a function h↦Dh which takes in an instance of the planar Gaussian free field and outputs a metric on the plane satisfying a certain list of natural axioms. We show that these axioms are satisfied for any subsequential limits of Liouville first passage percolation. Such subsequential limits were proven to exist by Ding et al. (Tightness of Liouville first passage percolation for γ∈(0, 2), 2019. ArXiv e-prints, arXiv:1904.08021). It is also known that these axioms are satisfied for the 8/3-LQG metric constructed by Miller and Sheffield (2013–2016). For any weak γ-LQG metric, we obtain moment bounds for diameters of sets as well as point-to-point, set-to-set, and point-to-set distances. We also show that any such metric is locally bi-Hölder continuous with respect to the Euclidean metric and compute the optimal Hölder exponents in both directions. Finally, we show that LQG geodesics cannot spend a long time near a straight line or the boundary of a metric ball. These results are used in subsequent work by Gwynne and Miller which proves that the weak γ-LQG metric is unique for each γ∈(0, 2), which in turn gives the uniqueness of the subsequential limit of Liouville first passage percolation. However, most of our results are new even in the special case when γ=8/3

Weak LQG metrics and Liouville first passage percolation

Funder: Columbia University Minerva fundAbstract: For γ∈(0, 2), we define a weakγ-Liouville quantum gravity (LQG) metric to be a function h↦Dh which takes in an instance of the planar Gaussian free field and outputs a metric on the plane satisfying a certain list of natural axioms. We show that these axioms are satisfied for any subsequential limits of Liouville first passage percolation. Such subsequential limits were proven to exist by Ding et al. (Tightness of Liouville first passage percolation for γ∈(0, 2), 2019. ArXiv e-prints, arXiv:1904.08021). It is also known that these axioms are satisfied for the 8/3-LQG metric constructed by Miller and Sheffield (2013–2016). For any weak γ-LQG metric, we obtain moment bounds for diameters of sets as well as point-to-point, set-to-set, and point-to-set distances. We also show that any such metric is locally bi-Hölder continuous with respect to the Euclidean metric and compute the optimal Hölder exponents in both directions. Finally, we show that LQG geodesics cannot spend a long time near a straight line or the boundary of a metric ball. These results are used in subsequent work by Gwynne and Miller which proves that the weak γ-LQG metric is unique for each γ∈(0, 2), which in turn gives the uniqueness of the subsequential limit of Liouville first passage percolation. However, most of our results are new even in the special case when γ=8/3

Asymptotic analysis of downlink MISO systems over Rician fading channels

International audienceIn this work, we focus on the ergodic sum rate in the downlink of a single-cell large-scale multiuser MIMO system in which the base station employs N antennas to communicate with K single-antenna user equipments. A regularized zero-forcing (RZF) scheme is used for precoding under the assumption that each link forms a spatially correlated MIMO Rician fading channel. The analysis is conducted assuming N and K grow large with a non trivial ratio and perfect channel state information is available at the base station. Recent results from random matrix theory and large system analysis are used to compute an asymptotic expression of the signal-to-interference-plus-noise ratio as a function of the system parameters, the spatial correlation matrix and the Rician factor. Numerical results are used to evaluate the performance gap in the finite system regime under different operating conditions