3,809 research outputs found
Modular forms, Schwarzian conditions, and symmetries of differential equations in physics
We give examples of infinite order rational transformations that leave linear
differential equations covariant. These examples are non-trivial yet simple
enough illustrations of exact representations of the renormalization group. We
first illustrate covariance properties on order-two linear differential
operators associated with identities relating the same hypergeometric
function with different rational pullbacks. We provide two new and more general
results of the previous covariance by rational functions: a new Heun function
example and a higher genus hypergeometric function example. We then
focus on identities relating the same hypergeometric function with two
different algebraic pullback transformations: such remarkable identities
correspond to modular forms, the algebraic transformations being solution of
another differentially algebraic Schwarzian equation that emerged in a paper by
Casale. Further, we show that the first differentially algebraic equation can
be seen as a subcase of the last Schwarzian differential condition, the
restriction corresponding to a factorization condition of some associated
order-two linear differential operator. Finally, we also explore
generalizations of these results, for instance, to , hypergeometric
functions, and show that one just reduces to the previous cases through
a Clausen identity.
In a hypergeometric framework the Schwarzian condition encapsulates
all the modular forms and modular equations of the theory of elliptic curves,
but these two conditions are actually richer than elliptic curves or
hypergeometric functions, as can be seen on the Heun and higher genus example.
This work is a strong incentive to develop more differentially algebraic
symmetry analysis in physics.Comment: 43 page
Scaling functions in the square Ising model
We show and give the linear differential operators of
order q= n^2/4+n+7/8+(-1)^n/8, for the integrals which appear in the
two-point correlation scaling function of Ising model . The integrals are given in expansion around r= 0 in the basis of the formal
solutions of with transcendental combination
coefficients. We find that the expression is a solution
of the Painlev\'e VI equation in the scaling limit. Combinations of the
(analytic at ) solutions of sum to .
We show that the expression is the scaling limit of the
correlation function and . The differential Galois
groups of the factors occurring in the operators are
given.Comment: 26 page
Diagonals of rational functions, pullbacked 2F1 hypergeometric functions and modular forms (unabrigded version)
We recall that diagonals of rational functions naturally occur in lattice
statistical mechanics and enumerative combinatorics. We find that a
seven-parameter rational function of three variables with a numerator equal to
one (reciprocal of a polynomial of degree two at most) can be expressed as a
pullbacked 2F1 hypergeometric function. This result can be seen as the simplest
non-trivial family of diagonals of rational functions. We focus on some
subcases such that the diagonals of the corresponding rational functions can be
written as a pullbacked 2F1 hypergeometric function with two possible rational
functions pullbacks algebraically related by modular equations, thus showing
explicitely that the diagonal is a modular form. We then generalise this result
to eight, nine and ten parameters families adding some selected cubic terms at
the denominator of the rational function defining the diagonal. We finally show
that each of these previous rational functions yields an infinite number of
rational functions whose diagonals are also pullbacked 2F1 hypergeometric
functions and modular forms.Comment: 39 page
Intermittency on catalysts
The present paper provides an overview of results obtained in four recent
papers by the authors. These papers address the problem of intermittency for
the Parabolic Anderson Model in a \emph{time-dependent random medium},
describing the evolution of a ``reactant'' in the presence of a ``catalyst''.
Three examples of catalysts are considered: (1) independent simple random
walks; (2) symmetric exclusion process; (3) symmetric voter model. The focus is
on the annealed Lyapunov exponents, i.e., the exponential growth rates of the
successive moments of the reactant. It turns out that these exponents exhibit
an interesting dependence on the dimension and on the diffusion constant.Comment: 11 pages, invited paper to appear in a Festschrift in honour of
Heinrich von Weizs\"acker, on the occasion of his 60th birthday, to be
published by Cambridge University Pres
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