Constructive Matrix Theory for Hermitian Higher Order Interaction

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Constructive Matrix Theory for Hermitian Higher Order Interaction
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2021
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In this seminar we study the constructive loop vertex expansion for stable matrix models with (single trace) interactions of arbitrarily high even order in the Hermitian and real symmetric cases. It relies on a new and simpler method which can also be applied in the previously treated complex case. We prove analyticity in the coupling constant of the free energy for such models in a domain uniform in the size of the matrix.
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[Music] [Music] the organizator of the seminar for and i think if inducted me i know it's a big risk they are taking because i speak very slowly i welcome welcome welcome casino not complete in english my accent will be awfully french and this seminar was originally scheduled in november 2019 in yeah i am very happy to give it all one year later constructive lobe matrices loop vertex matrices force a force table matrix model with single trace interaction of archipelage high even order in the hermitian case is a complete title it his joint work with thomas x crash scene and vasily saronos achieve 1910 dot dot 13 2561 we prove analyticity in the complicated as a cross being constant of the free energy to such models in a domain uniform in the size and of the matrix it is the groups of the theorem uniform in styles and it relies on new in and simpler method which can be also applied in the general case of norn mission matrix which was earlier treated by the same authors please i use you not to be shy not to be afraid of interrupting me independent and identically distributed measure with covariance of one of over n with a normalization and a gaussian behavior noted in the usual conventions that you can't see in the screen the at uh hp 2p model is defined by an action defined by s of lambda in a h equals lambda trace of h to the p p greater than three the seminar tehran in the case of p842 was treated much earlier it constituted the birth of the loop vertex is expansion now the partition function and a free energy of the model z of lambda and n equals sum over d mu of exponential minus n s of lambda and n and uh f of lambda n equals one over n squared a lot of and log on zed right uh k equal eight square root of uh one up loose along the h to the to be minus two so k square it's called a square plus lambda h to the 2p and put t because it's square of a k square square z times t z to the p minus t z a plus one equals zero with a z equals a minus lambda k to the 2 p minus 2. the sounds of the sounds of variable if that 2 to k h of k equals k square root of t of z um let us divine f of lambda f lambda of u equal square root of uh t of minus lambda u to the two p minus two hr lambda of u equals u times v f lambda of u and k lambda of v equals v times a square root of one plus lambda v 2 p minus 2 so that the function h and k are inverse of each other d h over decay equals as determinant
of h terms of one minus one transaction over uh k times over one this is a formula with is a true quest generally it can be checked through perturbation theory uh it's hold non-particularly as well uh borja somebody then it comes the main theorem of the paper theorem for and cylinders small enough so that the extension is absolutely convergent and defined an analytic function of lambda uniformly bounded in the uniform in n pacman domain defined uh by p of epsilon and eta equals the set of zero likes one modulus of lambda less than ita modulus of argument of lambda less than pi minus epsilon the main interest an expansion explicitly uh and absolutely convergent in the line of the loop vertex expansion now in the figure is is the pacman domain with a radius less than ita and angular size less upon 2 pi minus 2 epsilon we first we expand the partition function as an integral a sum of an integral of the vertices as in a perturbative theory then they apply the b k r a r formula the b k a r formula is a taylor expansion formula with integral reminder in several variables the result is a sum of the set tracks f of forest f on and labyled vertices of integral the main difficulty of this at first sight site indeed daunting formula is that this integral is computed of a parameter x with uh is the minimum of a set made of the parameter from a to b if there are a path from a to b in the forest and zero is if there is known this impressive format here is a picture of the first complete graph up to after seven now in example in the case and equal to there is a two forest including the empty one and the btar formula is a symbol simply f of one equal f of zero plus the sum of uh uh from zero up to one uh w times f prime of w uh we recognize as a formula at order one one with an integral argument and now [Music] take a another example for an equal three there are seven forests the it is there that the mean formula appears for the first time the formula is decomposed into the empty forest free sing tone forest one one zero zero zero one zero and zero zero one with a single parameter and two uh [Music] with two parameters one here is the list a one dupleton forest in example is a double shot from zero up to one one uh over w one w 2 times d square 1 2 of f f of w 1 w 2 mean of w 1 and w 2 and so on introducing seeing uh the condensed notation number that you are familiar with the bk a r formula and introducing the contour star notation the mu d d k for s and
s n equals the spot of the a going to one uh to n as f s of lambda we obtain uh some of the uh unlimited is apart from trivial factors an integral of a gaussian of the derivative completed at the value x of the forest the good thing is that the free energy f of lambda and n is computed by the same song of the same amplitude but made of spanning trees uh frasch n the statement if the partition function is made of some discrete object the logarithm the logarithm is made of the same object but restricted to the connected case is a true for a much finer class as and the class of graphs of the class of forest it pertains to the class of componentory species defined by andres wagon developed by canadian mathematician francois bergeron gila labelle and pierre leroux in a sense it is an ice cracked systematic method for containing district structure uh for example a merged of graphs of pagan mutations of metroid deal to alliance of cal poly models any schwimberger function s is expressed in a constructive way of the simply uh month of the participative series in the end if we get an expansion in terms of square root of lambda phi square sigma s is given by a sum of g t r in g of w of g g e h of g t and equal when rearranged as the sum of uh t of a of t a d of t equals uh sum of uh j super set of uh t of w of g t a of g it's a simple reactant with the sum of the modulus of a t being now convergent are we when the p weights are defined by the percentage of the f sector we have one of which the scrooge carl tree is living is you don't know what is a corrosion please ask in the set of question times accentment is a consecrative theory of transfer models versus matrix model he arises on until yearly on loop vertex expansion there are plenty of works via a lot of a lot of uh of authors a paper by lyrica johnny and myself is especially good for noticing defined an analytic theorem for transfers with positive single trace of a higher order in the figure two graphs are positive the first one does not symmetry axis of who is malonic and of the moment if we do not know if the bound is uniform along the expected domain optimal in n we do not even know what is what domain i give you some ideas of the proof of the theorem we fix the tree with at least and greater than two nodes the case and equal one requires a spatial treatment we put g of u equals h of u minus minus u notice that uh that z vanishes at lambda equals zero
so that g of u it was the third reminder of its derivative uh and now we've used factorization through allomorphic calculus f of h equals a circular integral over gamma of dv times f of ev over v minus h provided the contour gamma includes the full sp tomb of h are shown a keyhole contour gamma and circling the spectrum of h which for each armation and lies on the real axis here is a picture of the vertex with the sum of of its corner operators noted all of k uk uk plus one there is only one operator k per vertex this is due to the the fact that the formula b k a air depends only on the vertex therefore there is one only one agar vector base independently so operator o of k c k [Music] eb with values o equals one of the one plus sigma a b times a three terms of the type one of one u minus mu plus three terms with the two terms are symmetric of each other now calling a new a equals h of mu a sigma is defined by one plus plus sigma to the power minus one minus k of nu b of by a variable the upper left corner between this half age indicated by the cup symbol contains a free one of u minus k operators with index in disease k k a and k plus one uh uh the following lemmas are a bit technical and could give some explanation during the question time the taylor ham defense of a kiliman lima 1 [Music] absolute value of g lambda of a u less one constant times lambda is the two power one over four p square times or uh you can see in the screen the left next le mans are born limato [Music] le manfree lima 5 lima 4 l and lima 5. this is uh the proof is now complete it is worth of noticing that the the lve expresses the preen energy constructively it has a greater advantage of being a convergent some when part of particular techniques in phenomenon graphs fail the trick of involving the parameter for each corner of each vertices is the key of the simplification made by this theorem in simplifies also the non-er mission complex case of the case of symmetric or symplectic matrices as was said the case and equals one requires a special treatment it is actually quite simple it evolved five term each with a different structure i could give again some explanation during the question time thank you for your passions thank you dear song are there questions
remarks or comments about your proofs thank you for giving a sketch of proofs
so we have
five minutes be before the okay no remark comment or uh is it uh is your talk preparing some tools for joseph bengalun joseph
is going to make some independent talks but the matrices it's on a ribbon graph and um joseph made some interesting conjectures [Music] having a [Music] general lies the v-bond graphs colored ribbon graphs it's there is a link to cancer graphs with the uh leukemia talk work of myself not the matrix talk
okay okay thank you dear uh vanco and uh i think we can have a little break until precisely joseph's talk at the 15.
it's a quite a delayed by sanjay talk in the journal club journal club
it is
two talks called collide almost okay
thank you for this [Music] pointer so i propose that we have a break until 15 20. are we okay and then
we resume it precisely at 15 20 but of course as they are chats they share you can chat together i have something to print in the meanwhile thank you very much [Music]
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