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Narrow escape theory deals with the first passage of a particle diffusing in a cavity with small circular windows on the cavity wall to one of the windows. Assuming that (i) the cavity has no size anisotropy and (ii) all windows are sufficiently far away from each other, the theory provides an analytical expression for the particle mean first-passage time (MFPT) to one of the windows. This expression shows that the MFPT depends on the only global parameter of the cavity, its volume, independent of the cavity shape, and is inversely proportional to the product of the particle diffusivity and the sum of the window radii. Amazing simplicity and universality of this result raises the question of the range of its applicability. To shed some light on this issue, we study the narrow escape problem in a cylindrical cavity of arbitrary size anisotropy with two small windows arbitrarily located on the cavity side wall. We derive an approximate analytical solution for the MFPT, which smoothly goes from the conventional narrow escape solution in an isotropic cavity when the windows are sufficiently far away from each other to a qualitatively different solution in a long cylindrical cavity (the cavity length significantly exceeds its radius). Our solution demonstrates the mutual influence of the windows on the MFPT and shows how it depends on the inter-window distance. A key step in finding the solution is an approximate replacement of the initial three-dimensional problem by an equivalent one-dimensional one, where the particle diffuses along the cavity axis and the small absorbing windows are modeled by delta-function sinks. Brownian dynamics simulations are used to establish the range of applicability of our approximate approach and to learn what it means that the two windows are far away from each other. [ABSTRACT FROM AUTHOR]

We consider the spectral radius of a large random matrix X with independent, identically distributed entries. We show that its typical size is given by a precise three-term asymptotics with an optimal error term beyond the radius of the celebrated circular law. The coefficients in this asymptotics are universal but they differ from a similar asymptotics recently proved for the rightmost eigenvalue of X in Cipolloni et al., Ann. Probab. 51(6), 2192–2242 (2023). To access the more complicated spectral radius, we need to establish a new decorrelation mechanism for the low-lying singular values of X − z for different complex shift parameters z using the Dyson Brownian Motion. [ABSTRACT FROM AUTHOR]

The interface tends to behave as a circular shape under the effect of surface tension in the two-dimensional two-phase fluid flow. In this study, based on the coupled volume-of-fluid and level set (VOSET) method, an accurate circle-based interface reconstruction (CIR) algorithm for structured meshes is proposed, which features a straightforward implementation procedure. A portion of the standard circle is employed to approximate the curve interface, without the need for complex classification, coordinate transformation, and equation transformation. The radius of this circle is computed simply by the curvature and signed distance at the mixed cell center, and the center of this circle is determined using a straightforward dichotomy method under the mass conservation constraints. Results indicate that the coupled VOSET and CIR (VOSET-CIR) method maintains superior computational accuracy in the signed distance, interface curvature, and dynamic interface reconstruction compared to the coupled VOSET and piecewise linear interface calculation (VOSET-PLIC) method, with accuracy improvements ranging from 34% to 1490% across different test cases. Furthermore, the VOSET-CIR method outperforms the efficient least squares volume-of-fluid interface reconstruction (ELVIRA) and PLIC algorithms in reconstructing random circles and surpasses the quadratic spline based interface reconstruction (QUASI) algorithm in reconstructing random squares. In terms of computational efficiency, except for the initial PLIC algorithm, the time costs of the VOSET-CIR method are substantially lower than those of the ELVIRA and QUASI algorithms, as well as the VOSET-PLIC method. [ABSTRACT FROM AUTHOR]