‌‌Özlem Tugfe Demir (KTH Royal Institute of Technology)

The Bussgang Decomposition of Nonlinear Communication Systems: Basic theory, Vector Extension, and Common Misconceptions

Many of the wireless communication systems are nonlinear due to, for example, hardware impairments, such as nonlinear amplifiers and finite-resolution quantisation. The Bussgang decomposition is a commonly used tool when analysing the performance of such systems. The decomposition provides an exact probabilistic relationship between the output and the input of a nonlinearity: the output is equal to a scaled version of the input plus uncorrelated distortion. The decomposition can be used to compute either exact performance results or lower bounds, where the uncorrelated distortion is treated as independent noise. In this talk, the basic theory will be explained, some key examples will be provided, and the theory will be extended to the complex-valued vector signals. In addition, common misconceptions regarding the application of Bussgang decomposition will be clarified.

‌Ioana Dumitriu (University of California)

Spectral Gap in Random (Bi)Regular Graphs and Hypergraphs

Random graphs and hypergraphs have been used for decades to model large-scale networks, from biological, to electrical, and to social. Various random graphs (and their not-so-random properties) have been connected to algorithms solving problems from community detection to matrix completion, coding theory, and various other statistics / machine learning fundamental questions; in the past decade, this research area has expanded to include random hypergraphs. One of these special properties is the spectral gap for graph-associated matrices; roughly speaking, it means that the main eigenvalue(s) are well-separated from the bulk and it guarantees strong connectivity properties. This talk will take a look at the spectra of adjacency / Laplacian matrices for some random regular models, explain how we know that the spectral gap is there, and connect spectral properties to the aforementioned applications. It will cover joint work with Gerandy Brito, Kameron Decker Harris, and Yizhe Zhu.

Maximilien Gadouleau  (Durham University)

Introduction to Lattice-based cryptography and its probabilistic aspects

Learning With Errors (LWE) and its variants is used in Lattice-based cryptography, one of the main contestants to become the next post-quantum standard for public-key cryptography. Unlike other cryptosystems, the encryption is based on potentially unbounded noise, rendering the decryption probabilistic, with a relatively small probability of decryption failure. Unfortunately, these decryption failures are a security weakness, hence different techniques to reduce the decryption error probability have been proposed. In this talk, we give an introduction to LWE and its variants. We then go through a general framework to evaluate and reduce the decryption error probability. This finally leads us to propose new parameters for some LWE-based cryptosystems. This is joint work with Karl Southern and Lawrence Mitchell.

Camille Male (Université de Bordeaux)

Freeness over the diagonal and CLT for independent Wigner matrices

We study the limiting covariance in the CLT for (unnormalised) traces of polynomials in independent Wigner and deterministic matrices. For unitary invariant random matrices, Mingo and Speicher’s notion of second-order freeness gives a universal rule to compute the covariance. But this one is in general not valid for non-Gaussian Wigner matrices, since the global fluctuations are not universal, depending in particular on the moment of order 4 of the Wigner matrices. Yet, we prove that for certain independent Wigner and deterministic matrices, the covariance actually follows a universal rule in the context of freeness over the diagonal.

Ralf Müller (Friedrich-Alexander-Universität)

Probability and the Complexity of Functions

The computation of complex functions is increasingly delegated to deep neural networks. In that way, the complexity of high-dimensional functions is reduced to matrix-vector products. Based on the properties of large random matrices with exponential aspect ratio, an algorithm is proposed to reduce the number of elementary computations in matrix-vector products from O(K2) to O(K2/log K). This is asymptotically inferior to the Strassen or Coppersmith Winograd algorithm for matrix-matrix multiplication. However, the new algorithm outperforms competing methods for the practically most relevant matrix sizes, i.e. several dozens to some thousands. Furthermore, the new algorithm computes matrix-vector products one-by-one, not only matrix-matrix products, which is important for real-time applications. Finally, an outlook is given, how to further reduce the complexity of high-dimensional functions, if the new algorithm is jointly applied across various layers of deep neural networks utilizing the Hausdorff metric.

Alessandra Occelli (University of Lisbon) ‌

Space-time covariance of KPZ growth models

We study the time-time covariance of KPZ growth processes in (1+1)-dimensions for generic initial conditions. We prove convergence of the covariance and local universality of the first order correction when the two observation times are close. Joint work with Patrik Ferrari.

Alperen Ozdemir (Georgia Institute of Technology)

Martingale decomposition Eulerian statistics

We decompose Eulerian statistics into martingale differences by viewing recursive progressions of arrays of numbers as random processes. This provides a new proof of the fact that the number of descents in random permutations is asymptotically normal with an error bound of order n-1/2. Similar results are obtained for other descent-related statistics including the number of inversions, descents in Stirling permutations, descents in involutions, descents in derangements, the length of the longest alternating subsequences and two-sided Eulerian numbers.

Jeremy Quastel (University of Toronto)

The KPZ Fixed Point

The 1-d KPZ universality class contains random interface growth models as well as random polymer free energies and driven diffusive systems. The strong coupling fixed point has been computed through the exact solution of a special model in the class, TASEP. It is a completely integrable Markov process: the transition probabilities being described by classical integrable PDE’s. The Tracy-Widom distributions arise as special self-similar solutions. The talk will be a gentle review of these developments.

‌Fredrik Viklund (KTH Royal Institute of Technology)

The Loewner-Kufarev energy and Weil-Petersson foliations

The Loewner-Kufarev equation takes as input a suitable family of measures on the unit circle and outputs a monotone family of simply connected domains by describing the dynamics of the corresponding family of Riemann maps. Many interesting planar growth processes, including (deterministic) Laplacian growth type processes and (random) SLEs, can be formulated within this framework.

I will discuss Loewner-Kufarev evolution corresponding to a particular family of (deterministic) measures defined by the condition to have finite Loewner-Kufarev energy, a quantity that arises in the context of large deviations of SLE processes when the kappa parameter is large. The evolving interfaces form a ``foliating’’ family of Weil-Petersson quasicircles, a class of (non-smooth) Jordan curves that in the last decades has received significant attention by both physicists and mathematicians, but only very recently by probabilists. These processes exhibit several interesting features and surprising symmetries linked to random conformal geometry. This is based on joint work with Yilin Wang (MIT).

‌Pierre Youssef (New York University in Abu Dhabi)

On the spectrum of random graphs

Understanding the distribution of the spectrum as the dimension grows is one of the main problems in random matrix theory. This includes, among others, the study of the limiting spectral distribution and the behaviour at the boundary of the support of the limiting measure. It is known that the empirical spectral distribution of a square random matrix (resp. symmetric) with i.i.d centred entries with unit variance converges to the circular law (resp. semi-circular) as the dimension grows. In this talk, we are interested in the stability of these results and the behaviour of the spectrum when the i.i.d assumption is relaxed. Random graphs provide models encapsulating sparsity and dependence.

The talk will investigate

1. The limiting spectral distribution of random regular graphs,
2. The behaviour of the extreme eigenvalues/singular values and the spectral gap of random graphs.