Abstract. The signed loop approach is a beautiful way to rigorously study the two-dimensional Ising model with no external field. In this paper, we explore the foundations of the method, including details that have so far been neglected or overlooked in the literature. We demonstrate how the method can be applied to the Ising model on the square lattice to derive explicit formal expressions for the free energy density and two-point functions in terms of sums over loops, valid all the way up to the self-dual point. As a corollary, it follows that the self-dual point is critical both for the behaviour of the free energy density, and for the decay of the two-point functions.

A matrix is called totally positive (resp. totally nonnegative) if all its minors are positive (resp. nonnegative). Consider the Ising model with free boundary conditions and no external field on a planar graph G. Let a 1 , . . . , a k , b k , . . . , b 1 be vertices placed in a counterclockwise order on the outer face of G. We show that the k × k matrix of the twopoint spin correlation functionsis totally nonnegative. Moreover, det M > 0 if and only if there exist k pairwise vertexdisjoint paths that connect a i with b i . We also compute the scaling limit at criticality of the probability that there are k parallel and disjoint connections between a i and b i in the double random current model. Our results are based on a new distributional relation between double random currents and random alternating flows of Talaska [37].

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Abstract. We provide an upper bound on the spectral radius of the Kac-Ward transition matrix for a general planar graph. Combined with the Kac-Ward formula for the partition function of the planar Ising model, this allows us to identify regions in the complex plane where the free energy density limits are analytic functions of the inverse temperature. The bound turns out to be optimal in the case of isoradial graphs, i.e. it yields criticality of the self-dual Z-invariant coupling constants.

We study spin systems defined by the winding of a random walk loop soup. For a particular choice of loop soup intensity, we show that the corresponding spin system is reflection-positive and is dual, in the Kramers-Wannier sense, to the spin system sgn(ϕ) where ϕ is a discrete Gaussian free field.In general, we show that the spin correlation functions have conformally covariant scaling limits corresponding to the one-parameter family of functions studied by Camia, Gandolfi and Kleban (Nuclear Physics B 902, 2016) and defined in terms of the winding of the Brownian loop soup. These functions have properties consistent with the behavior of correlation functions of conformal primaries in a conformal field theory. Here, we prove that they do correspond to correlation functions of continuum fields (random generalized functions) for values of the intensity of the Brownian loop soup that are not too large.2010 Mathematics Subject Classification. 82B20, 60G60, 60G18, 82B41.

The random walk loop soup is a Poissonian ensemble of lattice loops; it has been extensively studied because of its connections to the discrete Gaussian free field, but was originally introduced by Lawler and Trujillo Ferreras as a discrete version of the Brownian loop soup of Lawler and Werner, a conformally invariant Poissonian ensemble of planar loops with deep connections to conformal loop ensembles (CLEs) and the Schramm-Loewner evolution (SLE). Lawler and Trujillo Ferreras showed that, roughly speaking, in the continuum scaling limit, "large" lattice loops from the random walk loop soup converge to "large" loops from the Brownian loop soup. Their results, however, do not extend to clusters of loops, which are interesting because the connection between Brownian loop soup and CLE goes via cluster boundaries. In this paper, we study the scaling limit of clusters of "large" lattice loops, showing that they converge to Brownian loop soup clusters. In particular, our results imply that the collection of outer boundaries of outermost clusters composed of "large" lattice loops converges to CLE. B Tim van de Brug

Abstract. We show that the critical Kac-Ward operator on isoradial graphs acts in a certain sense as the operator of s-holomorphicity, and we identify the fermionic observable for the spin Ising model as the inverse of this operator. This result is partially a consequence of a more general observation that the inverse Kac-Ward operator on any planar graph is given by what we call a fermionic generating function. Furthermore, using bounds obtained in [20] for the spectral radius and operator norm of the Kac-Ward transition matrix, we provide a general picture of the non-backtracking walk representation of the critical and supercritical inverse Kac-Ward operators on isoradial graphs.

Staphylococcus aureus is a commensal inhabitant of skin and mucous membranes in nose vestibule but also an important opportunistic pathogen of humans and livestock. The extracellular proteome as a whole constitutes its major virulence determinant; however, the involvement of particular proteins is still relatively poorly understood. In this study, we compared the extracellular proteomes of poultry-derived S. aureus strains exhibiting a virulent (VIR) and non-virulent (NVIR) phenotype in a chicken embryo experimental infection model with the aim to identify proteomic signatures associated with the particular phenotypes. Despite significant heterogeneity within the analyzed proteomes, we identified alpha-haemolysin and bifunctional autolysin as indicators of virulence, whereas glutamylendopeptidase production was characteristic for non-virulent strains. Staphopain C (StpC) was identified in both the VIR and NVIR proteomes and the latter fact contradicted previous findings suggesting its involvement in virulence. By supplementing NVIR, StpC-negative strains with StpC, and comparing the virulence of parental and supplemented strains, we demonstrated that staphopain C alone does not affect staphylococcal virulence in a chicken embryo model.