Let $p(Y_1, dots, Y_d, Z_1, dots, Z_e)$ be a self-adjoint noncommutative polynomial, with coefficients from $mathbb{C}^{r times r}$, in the indeterminates $Y_1, dots, Y_d$ (considered to be self-adjoint), the indeterminates $Z_1, dots, Z_e$, and thei
r adjoints $Z_1^*, dots, Z_e^*$. Suppose $Y_1, dots, Y_d$ are replaced by independent random $n times n$ matching matrices, and $Z_1, dots, Z_e$ are replaced by independent random $n times n$ permutation matrices. Assuming for simplicity that $p$s coefficients are $0$-$1$ matrices, the result can be thought of as a kind of random $rn$-vertex graph $G$. As $n to infty$, there will be a natural limiting infinite graph $X$ that covers any finite outcome for $G$. A recent landmark result of Bordenave and Collins shows that for any $varepsilon > 0$, with high probability the spectrum of a random $G$ will be $varepsilon$-close in Hausdorff distance to the spectrum of $X$ (once the suitably defined trivial eigenvalues are excluded). We say that $G$ is $varepsilon$-near fully $X$-Ramanujan. Our work has two contributions: First we study and clarify the class of infinite graphs $X$ that can arise in this way. Second, we derandomize the Bordenave-Collins result: for any $X$, we provide explicit, arbitrarily large graphs $G$ that are covered by $X$ and that have (nontrivial) spectrum at Hausdorff distance at most $varepsilon$ from that of $X$. This significantly generalizes the recent work of Mohanty et al., which provided explicit near-Ramanujan graphs for every degree $d$ (meaning $d$-regular graphs with all nontrivial eigenvalues bounded in magnitude by $2sqrt{d-1} + varepsilon$). As an application of our main technical theorem, we are also able to determine the eigenvalue relaxation value for a wide class of average-case degree-$2$ constraint satisfaction problems.
The phase transitions from one plateau to the next plateau or to an insulator in quantum Hall and quantum anomalous Hall (QAH) systems have revealed universal scaling behaviors. A magnetic-field-driven quantum phase transition from a QAH insulator to
an axion insulator was recently demonstrated in magnetic topological insulator sandwich samples. Here, we show that the temperature dependence of the derivative of the longitudinal resistance on magnetic field at the transition point follows a characteristic power-law that indicates a universal scaling behavior for the QAH to axion insulator phase transition. Similar to the quantum Hall plateau to plateau transition, the QAH to axion insulator transition can also be understood by the Chalker-Coddington network model. We extract a critical exponent k~ 0.38 in agreement with recent high-precision numerical results on the correlation length exponent of the Chalker-Coddington model at v ~ 2.6, rather than the generally-accepted value of 2.33.
This paper concerns the dynamics of a layer of incompressible viscous fluid lying above a vertically oscillating rigid plane and with an upper boundary given by a free surface. We consider the problem with gravity and surface tension for horizontally
periodic flows. This problem gives rise to flat but vertically oscillating equilibrium solutions, and the main thrust of this paper is to study the asymptotic stability of these equilibria in certain parameter regimes. We prove that both with and without surface tension there exists a parameter regime in which sufficiently small perturbations of the equilibrium at time $t = 0$ give rise to global-in-time solutions that decay to equilibrium at an identified quantitative rate.
