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1. The second Vassiliev measure of uniform random walks and polygons in confined space

   https://doi.org/10.1088/1751-8121/ac4abf  

   Smith, Philip; Panagiotou, Eleni (February 2022, Journal of Physics A: Mathematical and Theoretical)

   Abstract Biopolymers, like chromatin, are often confined in small volumes. Confinement has a great effect on polymer conformations, including polymer entanglement. Polymer chains and other filamentous structures can be represented by polygonal curves in three-space. In this manuscript, we examine the topological complexity of polygonal chains in three-space and in confinement as a function of their length. We model polygonal chains by equilateral random walks in three-space and by uniform random walks (URWs) in confinement. For the topological characterization, we use the second Vassiliev measure. This is an integer topological invariant for polygons and a continuous functions over the real numbers, as a function of the chain coordinates for open polygonal chains. For URWs in confined space, we prove that the average value of the Vassiliev measure in the space of configurations increases as O ( n 2 ) with the length of the walks or polygons. We verify this result numerically and our numerical results also show that the mean value of the second Vassiliev measure of equilateral random walks in three-space increases as O ( n ). These results reveal the rate at which knotting of open curves and not simply entanglement are affected by confinement. 

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2. Signed area enumeration for lattice paths

   Polychronakos, Alexios P; Ouvry, Stephane (April 2023, Séminaire lotharingien de combinatoire)

   We give a summary of recent progress on the signed area enumeration of 9 closed walks on planar lattices. Several connections are made with quantum mechanics 10 and statistical mechanics. Explicit combinatorial formulae are proposed which rely on 11 sums labelled by the multicompositions of the length of the walks. 

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3. Signed area enumeration for lattice paths

   Stephane, Ouvry; Polychronakos, Alexios P. (April 2023, Séminaire lotharingien de combinatoire)

   We give a summary of recent progress on the signed area enumeration of 9 closed walks on planar lattices. Several connections are made with quantum mechanics 10 and statistical mechanics. Explicit combinatorial formulae are proposed which rely on 11 sums labelled by the multicompositions of the length of the walks. 

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4. Signed area enumeration for lattice paths

   Ouvry, Stephane and (April 2023, Séminaire lotharingien de combinatoire)

   We give a summary of recent progress on the signed area enumeration of 9 closed walks on planar lattices. Several connections are made with quantum mechanics 10 and statistical mechanics. Explicit combinatorial formulae are proposed which rely on 11 sums labelled by the multicompositions of the length of the walks. 

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5. Least squares as random walks

   https://doi.org/10.1016/j.physleta.2025.130449  

   Kostinski, Alexander; Ierley, Glenn; Kostinski, Sarah (June 2025, Physics Letters A)

   Linear least squares (LLS) is perhaps the most common method of data analysis, dating back to Legendre, Gauss and Laplace. Framed as linear regression, LLS is also a backbone of mathematical statistics. Here we report on an unexpected new connection between LLS and random walks. To that end, we introduce the notion of a random walk based on a discrete sequence of data samples (data walk). We show that the slope of a straight line which annuls the net area under a residual data walk equals the one found by LLS. For equidistant data samples this result is exact and holds for an arbitrary distribution of steps. 

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