The free multiplicative Brownian motion
The proximity of many strongly correlated superconductors to density-wave or nematic order has led to an extensive search for fingerprints of pairing mediated by dynamical quantum-critical (QC) fluctuations of the corresponding order parameter. Here we study anisotropic
- Publication Date:
- NSF-PAR ID:
- 10154251
- Journal Name:
- npj Quantum Materials
- Volume:
- 4
- Issue:
- 1
- ISSN:
- 2397-4648
- Publisher:
- Nature Publishing Group
- Sponsoring Org:
- National Science Foundation
More Like this
-
Abstract is the large-$$b_{t}$$ N limit of the Brownian motion on in the sense of$$\mathsf {GL}(N;\mathbb {C}),$$ -distributions. The natural candidate for the large-$$*$$ N limit of the empirical distribution of eigenvalues is thus the Brown measure of . In previous work, the second and third authors showed that this Brown measure is supported in the closure of a region$$b_{t}$$ that appeared in the work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous density$$\Sigma _{t}$$ on$$W_{t}$$ which is strictly positive and real analytic on$$\overline{\Sigma }_{t},$$ . This density has a simple form in polar coordinates:$$\Sigma _{t}$$ where$$\begin{aligned} W_{t}(r,\theta )=\frac{1}{r^{2}}w_{t}(\theta ), \end{aligned}$$ is an analytic function determined by the geometry of the region$$w_{t}$$ . We show also that the spectral measure of free unitary Brownian motion$$\Sigma _{t}$$ is a “shadow” of the Brown measure of$$u_{t}$$ , precisely mirroring the relationship between the circular and semicircular laws. We develop several new methods, based on stochastic differential equations and PDE, to prove these results.$$b_{t}$$ -
Abstract It has been recently established in David and Mayboroda (Approximation of green functions and domains with uniformly rectifiable boundaries of all dimensions.
arXiv:2010.09793 ) that on uniformly rectifiable sets the Green function is almost affine in the weak sense, and moreover, in some scenarios such Green function estimates are equivalent to the uniform rectifiability of a set. The present paper tackles a strong analogue of these results, starting with the “flagship degenerate operators on sets with lower dimensional boundaries. We consider the elliptic operators associated to a domain$$L_{\beta ,\gamma } =- {\text {div}}D^{d+1+\gamma -n} \nabla $$ with a uniformly rectifiable boundary$$\Omega \subset {\mathbb {R}}^n$$ of dimension$$\Gamma $$ , the now usual distance to the boundary$$d < n-1$$ given by$$D = D_\beta $$ for$$D_\beta (X)^{-\beta } = \int _{\Gamma } |X-y|^{-d-\beta } d\sigma (y)$$ , where$$X \in \Omega $$ and$$\beta >0$$ . In this paper we show that the Green function$$\gamma \in (-1,1)$$ G for , with pole at infinity, is well approximated by multiples of$$L_{\beta ,\gamma }$$ , in the sense that the function$$D^{1-\gamma }$$ satisfies a Carleson measure estimate on$$\big | D\nabla \big (\ln \big ( \frac{G}{D^{1-\gamma }} \big )\big )\big |^2$$ . We underline that the strong and the weak results are different in nature and, of course, at the levelmore »$$\Omega $$ -
Abstract We consider a model of electrons at zero temperature, with a repulsive interaction which is a function of the energy transfer. Such an interaction can arise from the combination of electron–electron repulsion at high energies and the weaker electron–phonon attraction at low energies. As shown in previous works, superconductivity can develop despite the overall repulsion due to the energy dependence of the interaction, but the gap Δ(
ω ) must change sign at some (imaginary) frequencyω 0to counteract the repulsion. However, when the constant repulsive part of the interaction is increased, a quantum phase transition towards the normal state occurs. We show that, as the phase transition is approached, Δ andω 0must vanish in a correlated way such that . We discuss the behavior of phase fluctuations near this transition and show that the correlation between Δ(0) and$$1/| \log [{{\Delta }}(0)]| \sim {\omega }_{0}^{2}$$ ω 0locks the phase stiffness to a non-zero value. -
Abstract Sequence mappability is an important task in genome resequencing. In the (
k ,m )-mappability problem, for a given sequenceT of lengthn , the goal is to compute a table whosei th entry is the number of indices such that the length-$$j \ne i$$ m substrings ofT starting at positionsi andj have at mostk mismatches. Previous works on this problem focused on heuristics computing a rough approximation of the result or on the case of . We present several efficient algorithms for the general case of the problem. Our main result is an algorithm that, for$$k=1$$ , works in$$k=O(1)$$ space and, with high probability, in$$O(n)$$ time. Our algorithm requires a careful adaptation of the$$O(n \cdot \min \{m^k,\log ^k n\})$$ k -errata trees of Cole et al. [STOC 2004] to avoid multiple counting of pairs of substrings. Our technique can also be applied to solve the all-pairs Hamming distance problem introduced by Crochemore et al. [WABI 2017]. We further develop -time algorithms to compute$$O(n^2)$$ all (k ,m )-mappability tables for a fixedm and all or a fixed$$k\in \{0,\ldots ,m\}$$ k and all . Finally, we show that, for$$m\in \{k,\ldots ,n\}$$ , the ($$k,m = \Theta (\log n)$$ k ,m )-mappability problem cannot be solved in strongly subquadratic time unless the Strong Exponential Time Hypothesis fails. This is an improved and extended version of a paper presented at SPIRE 2018. -
Abstract Let
be an elliptically fibered$$X\rightarrow {{\mathbb {P}}}^1$$ K 3 surface, admitting a sequence of Ricci-flat metrics collapsing the fibers. Let$$\omega _{i}$$ V be a holomorphicSU (n ) bundle overX , stable with respect to . Given the corresponding sequence$$\omega _i$$ of Hermitian–Yang–Mills connections on$$\Xi _i$$ V , we prove that, ifE is a generic fiber, the restricted sequence converges to a flat connection$$\Xi _i|_{E}$$ . Furthermore, if the restriction$$A_0$$ is of the form$$V|_E$$ for$$\oplus _{j=1}^n{\mathcal {O}}_E(q_j-0)$$ n distinct points , then these points uniquely determine$$q_j\in E$$ .$$A_0$$