7,475 research outputs found
Yang-Lee Zeros of the Two- and Three-State Potts Model Defined on Feynman Diagrams
We present both analytic and numerical results on the position of the
partition function zeros on the complex magnetic field plane of the
(Ising) and states Potts model defined on Feynman diagrams
(thin random graphs). Our analytic results are based on the ideas of
destructive interference of coexisting phases and low temperature expansions.
For the case of the Ising model an argument based on a symmetry of the saddle
point equations leads us to a nonperturbative proof that the Yang-Lee zeros are
located on the unit circle, although no circle theorem is known in this case of
random graphs. For the states Potts model our perturbative results
indicate that the Yang-Lee zeros lie outside the unit circle. Both analytic
results are confirmed by finite lattice numerical calculations.Comment: 16 pages, 2 figures. Third version: the title was slightly changed.
To be published in Physical Review
Bounds for extreme zeros of some classical orthogonal polynomials
We derive upper bounds for the smallest zero and lower bounds for the largest
zero of Laguerre, Jacobi and Gegenbauer polynomials. Our approach uses mixed
three term recurrence relations satisfied by polynomials corresponding to
different parameter(s) within the same classical family. We prove that
interlacing properties of the zeros impose restrictions on the possible
location of common zeros of the polynomials involved and deduce strict bounds
for the extreme zeros of polynomials belonging to each of these three classical
families. We show numerically that the bounds generated by our method improve
known lower (upper) bounds for the largest (smallest) zeros of polynomials in
these families, notably in the case of Jacobi and Gegenbauer polynomials
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