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Register a new userGeneralized disconnection exponents
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Wei Qian
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We introduce and compute the generalized disconnection exponents $eta_kappa(beta)$ which depend on $kappain(0,4]$ and another real parameter $beta$, extending the Brownian disconnection exponents (corresponding to $kappa=8/3$) computed by Lawler, Schramm and Werner 2001 (conjectured by Duplantier and Kwon 1988). For $kappain(8/3,4]$, the generalized disconnection exponents have a physical interpretation in terms of planar Brownian loop-soups with intensity $cin (0,1]$, 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 $cin(0,1)$ and equal to zero for the critical intensity $c=1$, 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 $kappa$ and two additional parameters $mu, u$, that we call radial hypergeometric SLEs. This is a natural but substantial extension of the family of radial SLE$_kappa(rho)s$.
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In this note, we show that the Lyapunov exponents of mixed products of random truncated Haar unitary and complex Ginibre matrices are asymptotically given by equally spaced `picket-fence statistics. We discuss how these statistics should originate from the connection between random matrix products and multiplicative Brownian motion on $operatorname{GL}_n(mathbb{C})$, analogous to the connection between discrete random walks and ordinary Brownian motion. Our methods are based on contour integral formulas for products of classical matrix ensembles from integrable probability.
We consider the bond percolation problem on a transient weighted graph induced by the excursion sets of the Gaussian free field on the corresponding cable system. Owing to the continuity of this setup and the strong Markov property of the field on the one hand, and the links with potential theory for the associated diffusion on the other, we rigorously determine the behavior of various key quantities related to the (near-)critical regime for this model. In particular, our results apply in case the base graph is the three-dimensional cubic lattice. They unveil the values of the associated critical exponents, which are explicit but not mean-field and consistent with predictions from scaling theory below the upper-critical dimension.
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P.G. Grinevich L.D. Landau Institute forn Theoretical Physics
, Lomonosov Moscow State University
2015
We construct a Moutard-type transform for the generalized analytic functions. The first theorems and the first explicit examples in this connection are given.
We compute the fluctuation exponents for a solvable model of one-dimensional directed polymers in random environment in the intermediate regime. This regime corresponds to taking the inverse temperature to zero with the size of the system. The exponents satisfy the KPZ scaling relation and coincide with physical predictions. In the critical case, we recover the fluctuation exponents of the Cole-Hopf solution of the KPZ equation in equilibrium and close to equilibrium.
We consider the complex eigenvalues of a Wishart type random matrix model $X=X_1 X_2^*$, where two rectangular complex Ginibre matrices $X_{1,2}$ of size $Ntimes (N+ u)$ are correlated through a non-Hermiticity parameter $tauin[0,1]$. For general $ u=O(N)$ and $tau$ we obtain the global limiting density and its support, given by a shifted ellipse. It provides a non-Hermitian generalisation of the Marchenko-Pastur distribution, which is recovered at maximal correlation $X_1=X_2$ when $tau=1$. The square root of the complex Wishart eigenvalues, corresponding to the non-zero complex eigenvalues of the Dirac matrix $mathcal{D}=begin{pmatrix} 0 & X_1 X_2^* & 0 end{pmatrix},$ are supported in a domain parametrised by a quartic equation. It displays a lemniscate type transition at a critical value $tau_c,$ where the interior of the spectrum splits into two connected components. At multi-criticality we obtain the limiting local kernel given by the edge kernel of the Ginibre ensemble in squared variables. For the global statistics, we apply Frostmans equilibrium problem to the 2D Coulomb gas, whereas the local statistics follows from a saddle point analysis of the kernel of orthogonal Laguerre polynomials in the complex plane.
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