Session Q09: Heineman Prize, APS Prize Medal, Oppenheim Award; Prize Session

Prize Talk: Dannie Heineman Prize for Mathematical PhysicsTitle: The Lace expansion and Random Walk Representation in Statistical Mechanics

For a large class of classical ferromagnetic lattice spin systems that includes the Ising model, the φ^4 lattice euclidean field theory and O(N) models, the two point function has a Random Walk Representation as a sum over self-interacting walks. This a prolific source of correlation inequalities. Applications include Froehlich's 1982 proof that continuum limits of φ^4 are Gaussian in five or more dimensions and a short construction of continuum φ^4 in two and three dimensions. The random walk representation shows that self-avoiding walk is "part" of φ^4. A graphical expansion for self-avoiding walk is obtained by expanding the self-interaction, but has far too many graphs to be absolutely convergent. However, there exists a resummation that simultaneously removes infrared divergences and eliminates most of the graphs.  The resulting Lace Expansion is convergent for self-avoiding walk in five or more dimensions, and convergence implies that the end-to-end distance of the walk grows as the square root of the number of steps. Similar lace resummations are possible for many systems: in 1990 Hara and Slade derived a lace expansion for critical percolation which converges in high dimensions and proves that the critical exponents of percolation have mean field values.  Recently the random walk expansion was used to derive a lace expansion for critical φ^4 which converges in five dimensions if the coupling constant is small and convergence implies that the two point function of critical lattice φ^4 equals the two point function of the massless free field with a correction that decays more rapidly.

The φ^4 lace expansion does not converge in the critical dimension four because it does not renormalise the coupling constant. Is there a convergent resummation that includes coupling constant renormalisation?  Can lace expansions prove rotational invariance of continuum limits of lattice models?   

Prize Talk: Irwin Oppenheim Award: Spectra of Large Networks: Theory and Applications

The interactions between the constituents of complex systems, such as, neural networks, ecosystems, or financial networks, can be described with large, directed networks.  In recent years, mathematical methods have been developed to determine the spectral properties of large, directed graphs as a function of their topological properties.  These approaches extend beyond the extensively studied case of densely connected networks, and instead consider networks of finite connectivity, as they occur in real-world systems.  In this talk I will summarize some key results on the spectral analysis of large graphs, specifically focusing on the role of network topology and the properties of the edge weights.  Subsequently, we discuss the implications of these results for the dynamics of complex systems defined on large networks.       

Prize Talk: Irwin Oppenheim Award IIIDynamical systems on large networks with predator-prey interactions are stable and exhibit oscillations

We analyze the stability of linear dynamical systems defined on sparse, random graphs with predator-prey, competitive, and mutualistic interactions. These systems are aimed at modeling the stability of fixed points in large systems defined on complex networks, such as ecosystems consisting of a large number of species that interact through a food web. We develop an exact theory for the spectral distribution and the leading eigenvalue of the corresponding sparse Jacobian matrices. This theory reveals that the nature of local interactions has a strong influence on a system's stability. We show that, in general, linear dynamical systems defined on random graphs with a prescribed degree distribution of unbounded support are unstable if they are large enough, implying a tradeoff between stability and diversity. Remarkably, in contrast to the generic case, antagonistic systems that contain only interactions of the predator-prey type can be stable in the infinite size limit. This feature for antagonistic systems is accompanied by a peculiar oscillatory behavior of the dynamical response of the system after a perturbation, when the mean degree of the graph is small enough. Moreover, for antagonistic systems we also find that there exist a dynamical phase transition and critical mean degree above which the response becomes nonoscillatory.   

Prize talk: APS Medal for Exceptional Achievement in ResearchSpintronics!

For 500 years we lived in an analogue world where information was stored and preserved on paper, in books and paintings and in videotapes. Today we live in a world that has become digital in just a short space of time, barely more than a decade ago. This was made possible by advances in computing power, communications, and, most importantly, by our ability to store all information digitally in the cloud, in magnetic disk drives. I will briefly introduce spin-based magnetic sensors – spin-valves – that made this technological revolution possible and discuss two other major spin-based digital memory storage technologies, magnetic random-access memory and magnetic racetrack memory. The former, proposed initially in 1995, became a mainstream foundry technology in 2019, and the latter is on the path towards replacing magnetic disk drives with a solid state device with no moving parts that is orders of magnitude faster and energy efficient. All three spintronic technologies, spin-valves, magnetic random-access memory and racetrack memory, are formed from exquisitely, atomically engineered magnetic heterostructures that can be mass produced. The properties of these devices are determined and/or manipulated by spin-polarized electrical currents or spin currents that are generated from charge currents via various intrinsic or extrinsic phenomena. In this talk I review the development of the field of spintronics, its impact and my own contributions over the past 3 decades.