### Residence time in presence of moving defects and obstacles

We discuss the properties of the residence time in presence of moving defects or obstacles for a particle performing a one dimensional random walk. More precisely, for a particle conditioned to exit through the right endpoint, we measure the typical time needed to cross the entire lattice in presence of defects. We find explicit formulae for the residence time and discuss several models of moving obstacles. The presence of a stochastic updating rule for the motion of the obstacle smoothens the local residence time profiles found in the case of a static obstacle. We finally discuss connections with applicative problems, such as the pedestrian motion in presence of queues and the residence time of water flows in runoff ponds.

### Lie symmetry structure of nonlinear wave equations

We study Lie point symmetry structure of generalized nonlinear wave equations in the $1+n$-dimensional space-time.

### Renormalization in string-localized field theories: a microlocal analysis

Using methods of microlocal analysis, we prove that renormalization stays a pure ultraviolet problem in string-localized field theories, despite the weaker localization. Thus, power counting does not lose its significance as an indicator for renormalizability. Our proof puts the conjecture that the good ultraviolet behavior of string-localized fields improves renormalizability on safe mathematical ground. It also follows that the standard renormalization methods can be employed for string-localized field theories without any major adjustments.

### Frobenius manifolds on orbit spaces of non-reflection representations

We prove the orbit spaces of some non-reflection representations of finite groups posses Frobenius manifold structures.

### Resistance distance distribution in large sparse random graphs

We consider an Erdos-Renyi random graph consisting of N vertices connected by randomly and independently drawing an edge between every pair of them with probability c/N so that at N->infinity one obtains a graph of finite mean degree c. In this regime, we study the distribution of resistance distances between the vertices of this graph and develop an auxiliary field representation for this quantity in the spirit of statistical field theory. Using this representation, a saddle point evaluation of the resistance distance distribution is possible at N->infinity in terms of an 1/c expansion. The leading order of this expansion captures the results of numerical simulations very well down to rather small values of c; for example, it recovers the empirical distribution at c=4 or 6 with an overlap of around 90%. At large values of c, the distribution tends to a Gaussian of mean 2/c and standard deviation sqrt{2/c^3}. At small values of c, the distribution is skewed toward larger values, as captured by our saddle point analysis, and many fine features appear in addition to the main peak, including subleading peaks that can be traced back to resistance distances between vertices of specific low degrees and the rest of the graph. We develop a more refined saddle point scheme that extracts the corresponding degree-differentiated resistance distance distributions. We then use this approach to recover analytically the most apparent of the subleading peaks that originates from vertices of degree 1. Rather intuitively, this subleading peak turns out to be a copy of the main peak, shifted by one unit of resistance distance and scaled down by the probability for a vertex to have degree 1. We comment on a possible lack of smoothness in the true N->infinity distribution suggested by the numerics.

### Conformal invariance of double random currents and the XOR-Ising model II: tightness and properties in the discrete

This is the second of two papers devoted to the proof of conformal invariance of the critical double random current and the XOR-Ising model on the square lattice. More precisely, we show the convergence of loop ensembles obtained by taking the cluster boundaries in the sum of two independent currents both with free or wired boundary conditions, and in the XOR-Ising models with free and plus/plus boundary conditions. Therefore we establish Wilson's conjecture on the XOR-Ising model. The strategy, which to the best of our knowledge is different from previous proofs of conformal invariance, is based on the characterization of the scaling limit of these loop ensembles as certain local sets of the continuum Gaussian Free Field. In this paper, we derive crossing properties of the discrete models required to prove this characterization.

### High-dimensional near-critical percolation and the torus plateau

We consider percolation on $\mathbb{Z}^d$ and on the $d$-dimensional discrete torus, in dimensions $d \ge 11$ for the nearest-neighbour model and in dimensions $d>6$ for spread-out models. For $\mathbb{Z}^d$, we employ a wide range of techniques and previous results to prove that there exist positive constants $c$ and $C$ such that the slightly subcritical two-point function and one-arm probabilities satisfy $\mathbb{P}_{p_c-\varepsilon}(0 \leftrightarrow x) \leq \frac{C}{\|x\|^{d-2}} e^{-c\varepsilon^{1/2} \|x\|} \quad \text{ and } \quad \frac{c}{r^{2}} e^{-C \varepsilon^{1/2}r} \leq \mathbb{P}_{p_c-\varepsilon}\Bigl(0 \leftrightarrow \partial [-r,r]^d \Bigr) \leq \frac{C}{r^2} e^{-c \varepsilon^{1/2}r}.$ Using this, we prove that throughout the critical window the torus two-point function has a "plateau," meaning that it decays for small $x$ as $\|x\|^{-(d-2)}$ but for large $x$ is essentially constant and of order $V^{-2/3}$ where $V$ is the volume of the torus. The plateau for the two-point function leads immediately to a proof of the torus triangle condition, which is known to have many implications for the critical behaviour on the torus, and also leads to a proof that the critical values on the torus and on $\mathbb{Z}^d$ are separated by a multiple of $V^{-1/3}$. The torus triangle condition and the size of the separation of critical points have been proved previously, but our proofs are different and are direct consequences of the bound on the $\mathbb{Z}^d$ two-point function. In particular, we use results derived from the lace expansion on $\mathbb{Z}^d$, but in contrast to previous work on high-dimensional torus percolation we do not need or use a separate torus lace expansion.

### Conformal invariance of double random currents and the XOR-Ising model I: identification of the limit

This is the first of two papers devoted to the proof of conformal invariance of the critical double random current and the XOR-Ising models on the square lattice. More precisely, we show the convergence of loop ensembles obtained by taking the cluster boundaries in the sum of two independent currents with free and wired boundary conditions, and in the XOR-Ising models with free and plus/plus boundary conditions. Therefore we establish Wilson's conjecture on the XOR-Ising model. The strategy, which to the best of our knowledge is different from previous proofs of conformal invariance, is based on the characterization of the scaling limit of these loop ensembles as certain local sets of the Gaussian Free Field. In this paper, we identify uniquely the possible subsequential limits of the loop ensembles. Combined with the second paper, this completes the proof of conformal invariance.