We derive explicit estimates for the asymptotics of the first-passage function for random walks on free groups supported on the powers of standard generators, and use them to prove the singularity of the hitting measure for a similarly defined class of random walks on Fuchsian groups.

We study the landscapes of multi-species spherical spin glasses. Our results determine the phase boundary for annealed trivialization of the number of critical points and establish its equivalence with a quenched strong topological trivialization property. Namely, in the “trivial” regime, the number of critical points is constant, all are well conditioned, and all approximate critical points are close to a true critical point. As a consequence, we deduce that Langevin dynamics at sufficiently low temperature has logarithmic mixing time. Our approach begins with the Kac–Rice formula. We characterize the annealed trivialization phase by explicitly solving a suitable multidimensional variational problem, obtained by simplifying certain asymptotic determinant formulas from (Probab. Math. Phys. 3 (2022) 731–789; Ann. Inst. Henri Poincaré Probab. Stat. 60 (2024) 636–657). To obtain more precise quenched results, we develop general purpose techniques to avoid subexponential correction factors and show nonexistence of approximate critical points. Many of the results are new, even in the one-species case.

Consider the asymmetric nearest-neighbor exclusion process (ASEP) on Z with single particle drift γ>0, starting from a Bernoulli product invariant measure νρ with density ρ. It is known that the position XN of a tagged particle, say initially at the origin, at time N satisfies an a.s. law of large numbers 1NXN→γ(1−ρ) as N↑∞. In this context, we study the “typical” behavior of the tagged particle and “bulk” density evolution subject to “atypical” events {XN≥AN} or {XN≤AN} for A≠γ(1−ρ). We detail different structures, depending on whether A<0, 0≤A<γ(1−ρ), γ(1−ρ)<A<γ or A≥γ, under which these atypical events are achieved, and compute associated large deviation costs. Among our results is an “upper tail” large deviation principle in scale N for 1NXN.

We study the critical behavior for percolation on inhomogeneous random networks on n vertices, where the weights of the vertices follow a power-law distribution with exponent τ∈(2,3). Such networks, often referred to as scale-free networks, exhibit critical behavior when the percolation probability tends to zero at an appropriate rate, as n→∞. We identify the critical window for several scale-free random graph models, such as the Norros–Reittu model, Chung–Lu model and generalized random graphs. Surprisingly, there exists a finite time inside the critical window, after which we see a sudden emergence of a “tiny” giant component. This is a novel behavior, which is in contrast with the critical behavior in other known universality classes with τ∈(3,4) and τ>4. Precisely, for edge-retention probabilities πn=λn−(3−τ)/2, there is an explicitly computable λc>0 such that the critical window is of the form λ∈(0,λc), where the largest clusters have size of order nβ with β=(τ2−4τ+5)/[2(τ−1)]∈[ 2−1,12) and have nondegenerate scaling limits, while in the supercritical regime λ>λc, a unique “tiny giant” component of size Θ(n) emerges, and its size concentrates. For λ∈(0,λ c), the scaling limit of the maximum component sizes can be described in terms of components of a one-dimensional inhomogeneous percolation model on Z+ studied in a seminal work by Durrett and Kesten (In A Tribute to Paul Erdős (1990) 161–176 Cambridge Univ. Press). For λ>λc, we use a relation to general inhomogeneous random graphs, as studied by Bollobás, Janson and Riordan (Random Structures Algorithms 31 (2007) 3–122), to prove that the sudden emergence of the tiny giant is caused by a phase transition inside a smaller core of vertices of weight of order at least n.

The conformal loop ensemble (CLE) has two phases: for κ∈(8/3,4], the loops are simple and do not touch each other or the boundary; for κ∈(4,8), the loops are nonsimple and may touch each other and the boundary. For κ∈(4,8), we derive the probability that the loop surrounding a given point touches the domain boundary. We also obtain the law of the conformal radius of this loop seen from the given point conditioned on the loop touching the boundary or not, refining a result of Schramm–Sheffield–Wilson (Comm. Math. Phys. (2009) 288 43–53). As an application, we exactly evaluate the CLE counterpart of the nested-path exponent for the Fortuin–Kasteleyn (FK) random cluster model recently introduced by Song–Tan–Zhang–Jacobsen–Nienhuis–Deng (J. Phys. A 55 (2022) Paper No. 204002). This exponent describes the asymptotic behavior of the number of nested open paths in the open cluster containing the origin when the cluster is large. For Bernoulli percolation, which corresponds to κ=6, the exponent was derived recently in Song–Jacobsen–Nienhuis–Sportiello–Deng (2023) by a color switching argument. For κ≠6 and, in particular, for the FK-Ising case, our formula appears to be new. Our derivation begins with Sheffield’s construction of CLE from which the quantities of interest can be expressed by radial SLE. We solve the radial SLE problem using the coupling between SLE and Liouville quantum gravity, along with the exact solvability of Liouville conformal field theory.

We consider one-dimensional branching Brownian motion in a spatially random branching environment (BBMRE) and show that for almost every realisation of the environment, the distributions of the maximal particle of the BBMRE recentred around its median are tight as time evolves. This result is in stark contrast to the fact that the transition fronts in the solution to the randomised Fisher–Kolmogorov–Petrovskii–Piskunov (F-KPP) equation are, in general, not bounded uniformly in time. In particular, this highlights that—when compared to the settings of homogeneous branching Brownian motion and the F-KPP equation in a homogeneous environment—the introduction of a random environment leads to a much more intricate behaviour.

In the setting of lattice gauge theories with finite (possibly non-Abelian) gauge groups at weak coupling, we prove exponential decay of correlations for a wide class of gauge invariant functions, which in particular includes arbitrary functions of Wilson loop observables.

We study biased random walks on dynamical percolation on Zd. We establish a law of large numbers and an invariance principle for the random walk using regeneration times. Moreover, we verify that the Einstein relation holds, and we investigate the speed of the walk as a function of the bias. While for d=1 the speed is increasing, we show that, in general, this fails in dimension d≥2. As our main result, we establish two regimes of parameters, separated by an explicit critical curve such that the speed is either eventually strictly increasing or eventually strictly decreasing. This is in sharp contrast to the biased random walk on a static supercritical percolation cluster where the speed is known to be eventually zero.