TY  - JOUR
A1  - Casel, Katrin
A1  - Fischbeck, Philipp
A1  - Friedrich, Tobias
A1  - Göbel, Andreas
A1  - Lagodzinski, J. A. Gregor
T1  - Zeros and approximations of Holant polynomials on the complex plane
JF  - Computational complexity : CC
N2  - We present fully polynomial time approximation schemes for a broad class of Holant problems with complex edge weights, which we call Holant polynomials. We transform these problems into partition functions of abstract combinatorial structures known as polymers in statistical physics. Our method involves establishing zero-free regions for the partition functions of polymer models and using the most significant terms of the cluster expansion to approximate them. Results of our technique include new approximation and sampling algorithms for a diverse class of Holant polynomials in the low-temperature regime (i.e. small external field) and approximation algorithms for general Holant problems with small signature weights. Additionally, we give randomised approximation and sampling algorithms with faster running times for more restrictive classes. Finally, we improve the known zero-free regions for a perfect matching polynomial.
KW  - Holant problems
KW  - approximate counting
KW  - partition functions
KW  - graph
KW  - polynomials
Y1  - 2022
U6  - https://doi.org/10.1007/s00037-022-00226-5
SN  - 1016-3328
SN  - 1420-8954
VL  - 31
IS  - 2
PB  - Springer
CY  - Basel
ER  - 
TY  - JOUR
A1  - Fischer, Florian
A1  - Keller, Matthias
T1  - Riesz decompositions for Schrödinger operators on graphs
JF  - Journal of mathematical analysis and applications
N2  - We study superharmonic functions for Schrodinger operators on general weighted graphs. Specifically, we prove two decompositions which both go under the name Riesz decomposition in the literature. The first one decomposes a superharmonic function into a harmonic and a potential part. The second one decomposes a superharmonic function into a sum of superharmonic functions with certain upper bounds given by prescribed superharmonic functions. As application we show a Brelot type theorem.
KW  - Potential theory
KW  - Green's function
KW  - Schrödinger operator
KW  - Weighted
KW  - graph
KW  - Subcritical
KW  - Greatest harmonic minorant
Y1  - 2021
U6  - https://doi.org/10.1016/j.jmaa.2020.124674
SN  - 0022-247X
SN  - 1096-0813
VL  - 495
IS  - 1
PB  - Elsevier
CY  - Amsterdam
ER  -