Pierre Mergny's research while affiliated with École Polytechnique and other places
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Publications (10)
In this note we study the right large deviation of the top eigenvalue (or singular value) of the sum or product of two random matrices A and B as their dimensions goes to infinity. We consider a general framework containing the cases where A and/or B are taken from an invariant ensemble or are fixed diagonal matrices. We show that the tilting metho...
In this note, we study the high-temperature convolution introduced in Ref.\ \cite{mergny_cconv}, between two symmetric Bernoulli distributions. We give an analytical expression for both the Stieltjes transform and the density. This result provides the first non-trivial expression for the high-temperature convolution of two distributions and gives a...
In this note we study the right large deviation of the top eigenvalue (or singular value) of the sum or product of two random matrices $\mathbf{A}$ and $\mathbf{B}$ as their dimensions goes to infinity. The matrices $\mathbf{A}$ and $\mathbf{B}$ are each assumed to be taken from an invariant (or bi-invariant) ensemble with a confining potential wit...
We study the rank one Harish-Chandra-Itzykson-Zuber integral in the limit where \frac{N\beta}{2} \to c N β 2 → c , called the high-temperature regime and show that it can be used to construct a promising one-parameter interpolation family, with parameter c between the classical and the free convolution. This c-convolution has a simple interpretatio...
We study the probability of stability of a large complex system of size N within the framework of a generalized May model, which assumes a linear dynamics of each population size n i (with respect to its equilibrium value): d n i d t = − a i n i − T ∑ j J i j n j . The a i > 0’s are the intrinsic decay rates, J ij is a real symmetric ( N × N ) Gaus...
We study the probability of stability of a large complex system of size $N$ within the framework of a generalized May model, which assumes a linear dynamics of each population size $n_i$ (with respect to its equilibrium value): $ \frac{\mathrm{d}\, n_i}{\mathrm{d}t} = - a_i n_i - \sqrt{T} \sum_{j} J_{ij} n_j $. The $a_i>0$'s are the intrinsic decay...
We study the rank one Harish-Chandra-Itzykson-Zuber integral in the limit where $\frac{N \beta}{2} \to c $, called the high temperature regime and show that it can be used to construct a promising one-parameter interpolation, with parameter $c$ between the classical and the free convolution. This $c$-convolution has a simple interpretation in terms...
In this note, we study the asymptotic of spherical integrals, which are analytical extension in index of the normalized Schur polynomials for $\beta =2$ , and of Jack symmetric polynomials otherwise. Such integrals are the multiplicative counterparts of the Harish-Chandra-Itzykson-Zuber (HCIZ) integrals, whose asymptotic are given by the so-called...
Identifying protein-protein interactions is crucial for a systems-level understanding of the cell. Recently, algorithms based on inverse statistical physics, e.g., direct coupling analysis (DCA), have allowed to use evolutionarily related sequences to address two conceptually related inference tasks: finding pairs of interacting proteins and identi...
Identifying protein-protein interactions is crucial for a systems-level understanding of the cell. Recently, algorithms based on inverse statistical physics, e.g. Direct Coupling Analysis (DCA), have allowed to use evolutionarily related sequences to address two conceptually related inference tasks: finding pairs of interacting proteins, and identi...
Citations
... Our work fits into a recent lineage of papers, starting with [GH20] and [GM20], which establish large-deviations principles for random matrices using the method of tilting by spherical integrals. Papers in this line include [BG20,AGH21,McK21,GH22,Hus22,BGH20] in the math literature, and [Mai21,MP22] in the physics literature. Compared to these previous works, the main technical novelty in our work is the establishment of rigorous non-Gaussian results beyond the so-called x c threshold. ...
... A similar expression for the Stieltjes transform of the measure is found in the context of the Finite Free Convolution in [199] with = n 2 N ⇤ and f the negative Markov-Krein transform of µ. In the same way, the same expression is obtained in [200] in the case of the rank-one HCIZ integral interpolating between classical and free convolution. In this context, = c with c the interpolation parameter, and f the Markov-Krein transform of µ. ...
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... Similarly to the case of landscape studies which were instrumental to understand glassy dynamics in terms of local minima and metastable states, it would be very interesting to connect the properties of these unstable equilibria (more generally, of heteroclinic networks formed by them [68]) to the dynamical behavior. We envisage that invadable equilibria also play a role in the dynamics [69], and the calculation of their complexity is ongoing, as well as the generalization to inhomogeneous carrying capacities κ i [63,70,71]. ...
... (6)-which show these sequences are good dimers. We see that these probabilities are quite high, proving that most of the necessary information needed to model a good dimer can be captured by pairwise interactions and local biases, in agreement with some recent works [13,14]. In addition, the generated sequences are far from the old ones and have high diversity between each other, as it can be seen in Fig. 3 (bottom right panel) where we plot the distribution of the Hamming distance for each pair of sequence; hence we are sampling a different subset of the dimer space. ...
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