Speculative sampling is a popular technique for accelerating generation in Large Language Models whereby one samples candidate tokens using a fast draft model and accept/reject them based on the target model's distribution. While speculative sampling was previously limited to discrete sequences, we extend it to denoising diffusion models, which are state-of-the-art generative models for image, videos and protein generation. Our experiments demonstrate significant generation speedup on various denoising diffusion models, halving the number of function evaluations, while generating exact samples from the target model. We finally explain how this strategy can be also be used to accelerate simulation of Langevin diffusions.

Which graphs $G$ admit a percolating phase (i.e. $p_c(G)<1$)? This seemingly simple question is one of the most fundamental ones in percolation theory. A famous argument due to Peierls implies that if the number of minimal cutsets of size $n$ from a vertex to infinity in the graph grows at most exponentially in $n$, then $p_c(G)<1$. Our first theorem establishes the converse of this statement. This implies, for instance, that if a (uniformly) percolating phase exists, then a "strongly percolating��� one also does. In a second theorem, we show that if the simple random walk on the graph is uniformly transient, then the number of minimal cutsets is bounded exponentially (and in particular $p_c<1$). Both proofs rely on a probabilistic method that uses a random set to generate a random minimal cutset whose probability of taking any given value is lower bounded exponentially on its size.
Based on a joint work with Philip Easo and Vincent Tassion.

Models of two-dimensional random geometry are obtained as universal scaling limits in the Gromov-Hausdorff sense of large graphs embedded in the sphere. These models, which include the Brownian sphere, the Brownian disk and the Brownian plane, are also closely related to the quantum surfaces studied by Miller and Sheffield. We will present recent progress in the study of these random metric spaces. In particular we will discuss some remarkable properties of geodesics and mention some open problems.

Symmetry plays an important role in quantum systems: it can constrain the dynamics, give rise to selection rules, and provide computation methods in quantum computers. In recent years there are also new types of symmetries called generalized symmetries discovered in many quantum systems, including non-invertible symmetry and higher group symmetry. These lectures will be about symmetries in various quantum systems and their applications such as constraints on the low energy dynamics. Examples will be discussed in the lectures include quantum mechanics systems, gauge theories, lattice models, and the symmetry includes ordinary and higher form symmetry as well non-invertible symmetry.

Th-Cosmo-talk on April 10, 2025, at KCL room K3.11.

Title: Dark Matter as Topological Order.

Abstract: We propose that topological order can replace sterile neutrinos as dark matter candidates to cancel the Standard Model global gravitational anomalies. Standard Model (SM) with 15 Weyl fermions per family (lacking the 16th, the sterile right-handed neutrino nuR) suffers from mixed gauge-gravitational anomalies tied to baryon number plus or minus lepton number B+(-)L symmetry. Including nuR per family can cancel these anomalies, but when B+(-)L symmetry is preserved as discrete finite subgroups rather than a continuous U(1), the perturbative local anomalies become nonperturbative global anomalies. We systematically enumerate these gauge-gravitational global anomalies involving discrete B+(-)L that are enhanced from the fermion parity Z2F to Z2NF. The discreteness of B+(-)L is constrained by multi-fermion deformations beyond-the-SM and the family number Nf. Unlike the free quadratic nuR Majorana mass gap preserving the minimal Z2F, we explore novel scenarios canceling (B+(-)L)-gravitational anomalies while preserving the Z2NF discrete symmetries, featuring 4-dimensional interacting gapped topological orders or gapless sectors (e.g., conformal field theories). We propose symmetric anomalous sectors as quantum dark matter to cancel SM global anomalies. We find the uniqueness of the family number at Nf = 3, such that when the representation of Z2NF from the faithful B+L for baryons at both Nf and N equal to 3 is extended to the faithful Q + NcL for quarks at N = NcNf = 9, this symmetry extension ZNc=3 to ZNcNf =9 to ZNf =3 matches with the topological order dark matter construction. Key implications include: (1) a 5th force mediating between SM and dark matter via discrete B+(-)L gauge fields, (2) dark matter as topological order quantum matter with gapped anyon excitations at ends of extended defects, and (3) Ultra Unification and topological leptogenesis. [Based on arXiv:2502.21319, arXiv:2501.00607, arXiv:2412.21196, arXiv:2411.05786, arXiv:2012.15860, arXiv:2112.14765, arXiv:2204.08393, arXiv:2302.14862, arXiv:2312.14928].

Active filaments, such as kinesin propelled microtubules in gliding assay experiments, give rise to a plethora of active phases. In order to better understand which features of these phases are emergent and which exist at even the single filament level, we investigate the dynamics of individual active elastic filaments with chiral self-propulsion. To this end, we study the fully general time evolution of an overdamped plane curve and derive equations for the evolution of the curve���s shape and orientational characteristics. Applying this formalism to the specific case of an active elastic filament with chiral self-propulsion, we determine that sufficiently flexible filaments can exhibit stationary states with shape multi-stability which in turn gives rise to rotational dynamics. Further, the time-dependent evolution towards such steady states is highly nontrivial with both wave-like and diffusive characteristics available depending on the elastic properties of the system.