Brownian Loops, Layering Fields and Imaginary Gaussian Multiplicative Chaos

We study fields reminiscent of vertex operators built from the Brownian loop soup in the limit as the loop soup intensity tends to infinity. More precisely, following Camia et al. (Nucl Phys B 902:483–507, 2016), we take a (massless or massive) Brownian loop soup in a planar domain and assign a random sign to each loop. We then consider random fields defined by taking, at every point of the domain, the exponential of a purely imaginary constant times the sum of the signs associated to the loops that wind around that point. For domains conformally equivalent to a disk, the sum diverges logarithmically due to the small loops, but we show that a suitable renormalization procedure allows to define the fields in an appropriate Sobolev space. Subsequently, we let the intensity of the loop soup tend to infinity and prove that these vertex-like fields tend to a conformally covariant random field which can be expressed as an explicit functional of the imaginary Gaussian multiplicative chaos with covariance kernel given by the Brownian loop measure. Besides using properties of the Brownian loop soup and the Brownian loop measure, a main tool in our analysis is an explicit Wiener–Itô chaos expansion of linear functionals of vertex-like fields. Our methods apply to other variants of the model in which, for example, Brownian loops are replaced by disks.

Generalized disconnection exponents

We introduce and compute the generalized disconnection exponents$$\eta _\kappa (\beta )$$ η κ ( β ) which depend on $$\kappa \in (0,4]$$ κ ∈ ( 0 , 4 ] and another real parameter $$\beta $$ β , extending the Brownian disconnection exponents (corresponding to $$\kappa =8/3$$ κ = 8 / 3 ) computed by Lawler, Schramm and Werner (Acta Math 187(2):275–308, 2001; Acta Math 189(2):179–201, 2002) [conjectured by Duplantier and Kwon (Phys Rev Lett 61:2514–2517, 1988)]. For $$\kappa \in (8/3,4]$$ κ ∈ ( 8 / 3 , 4 ] , the generalized disconnection exponents have a physical interpretation in terms of planar Brownian loop-soups with intensity $$c\in (0,1]$$ c ∈ ( 0 , 1 ] , which allows us to obtain the first prediction of the dimension of multiple points on the cluster boundaries of these loop-soups. In particular, according to our prediction, the dimension of double points on the cluster boundaries is strictly positive for $$c\in (0,1)$$ c ∈ ( 0 , 1 ) and equal to zero for the critical intensity $$c=1$$ c = 1 , leading to an interesting open question of whether such points exist for the critical loop-soup. Our definition of the exponents is based on a certain general version of radial restriction measures that we construct and study. As an important tool, we introduce a new family of radial SLEs depending on $$\kappa $$ κ and two additional parameters $$\mu , \nu $$ μ , ν , that we call radial hypergeometric SLEs. This is a natural but substantial extension of the family of radial $$\hbox {SLE}_\kappa (\rho )s$$ SLE κ ( ρ ) s .

DNA-segment-capture model for loop extrusion by structural maintenance of chromosome (SMC) protein complexes

Cells possess remarkable control of the folding and entanglement topology of long and flexible chromosomal DNA molecules. It is thought that structural maintenance of chromosome (SMC) protein complexes play a crucial role in this, by organizing long DNAs into series of loops. Experimental data suggest that SMC complexes are able to translocate on DNA, as well as pull out lengths of DNA via a ‘loop extrusion’ process. We describe a Brownian loop-capture-ratchet model for translocation and loop extrusion based on known structural, catalytic, and DNA-binding properties of the Bacillus subtilis SMC complex. Our model provides an example of a new class of molecular motor where large conformational fluctuations of the motor ‘track’—in this case DNA—are involved in the basic translocation process. Quantitative analysis of our model leads to a series of predictions for the motor properties of SMC complexes, most strikingly a strong dependence of SMC translocation velocity and step size on tension in the DNA track that it is moving along, with ‘stalling’ occuring at subpiconewton tensions. We discuss how the same mechanism might be used by structurally related SMC complexes (Escherichia coli MukBEF and eukaryote condensin, cohesin and SMC5/6) to organize genomic DNA.