10,949 research outputs found
Inequalities for trace anomalies, length of the RG flow, distance between the fixed points and irreversibility
I discuss several issues about the irreversibility of the RG flow and the
trace anomalies c, a and a'. First I argue that in quantum field theory: i) the
scheme-invariant area Delta(a') of the graph of the effective beta function
between the fixed points defines the length of the RG flow; ii) the minimum of
Delta(a') in the space of flows connecting the same UV and IR fixed points
defines the (oriented) distance between the fixed points; iii) in even
dimensions, the distance between the fixed points is equal to
Delta(a)=a_UV-a_IR. In even dimensions, these statements imply the inequalities
0 =< Delta(a)=< Delta(a') and therefore the irreversibility of the RG flow.
Another consequence is the inequality a =< c for free scalars and fermions (but
not vectors), which can be checked explicitly. Secondly, I elaborate a more
general axiomatic set-up where irreversibility is defined as the statement that
there exist no pairs of non-trivial flows connecting interchanged UV and IR
fixed points. The axioms, based on the notions of length of the flow, oriented
distance between the fixed points and certain "oriented-triangle inequalities",
imply the irreversibility of the RG flow without a global a function. I
conjecture that the RG flow is irreversible also in odd dimensions (without a
global a function). In support of this, I check the axioms of irreversibility
in a class of d=3 theories where the RG flow is integrable at each order of the
large N expansion.Comment: 24 pages, 3 figures; expanded intro, improved presentation,
references added - CQ
Weighted Frechet Means as Convex Combinations in Metric Spaces: Properties and Generalized Median Inequalities
In this short note, we study the properties of the weighted Frechet mean as a
convex combination operator on an arbitrary metric space, (Y,d). We show that
this binary operator is commutative, non-associative, idempotent, invariant to
multiplication by a constant weight and possesses an identity element. We also
treat the properties of the weighted cumulative Frechet mean. These tools allow
us to derive several types of median inequalities for abstract metric spaces
that hold for both negative and positive Alexandrov spaces. In particular, we
show through an example that these bounds cannot be improved upon in general
metric spaces. For weighted Frechet means, however, such inequalities can
solely be derived for weights equal or greater than one. This latter limitation
highlights the inherent difficulties associated with working with
abstract-valued random variables.Comment: 7 pages, 1 figure. Submitted to Probability and Statistics Letter
Metric Cotype
We introduce the notion of metric cotype, a property of metric
spaces related to a property of normed spaces, called Rademacher
cotype. Apart from settling a long standing open problem in metric
geometry, this property is used to prove the following dichotomy: A
family of metric spaces F is either almost universal (i.e., contains
any finite metric space with any distortion > 1), or there exists
α > 0, and arbitrarily large n-point metrics whose distortion when
embedded in any member of F is at least Ω((log n)^α). The same
property is also used to prove strong non-embeddability theorems
of L_q into L_p, when q > max{2,p}. Finally we use metric cotype
to obtain a new type of isoperimetric inequality on the discrete
torus
- …