 constructive matrix theory for admission of your order interaction. I've hunked the organizer of the seminar for having invited me. I know it's a big risk they are taking because I speak very slowly. Welcome, welcome, welcome. Ma rédication for Afazie is not complete in English. My accent will be awfully French. And this seminar was originally scheduled in November 2019 in the EHL. I am very happy to give it all while you're at it. constructive loop matrices, loop vertex matrices, force table matrix model with single trace interaction of architecture is high. Even order is the Hermitian case is the complete title. It sees joint work with Thomas Krasche and Vasili Saronov. Archive, 1910, dovet, 13,261. We've proven analyticity in the complicated constant of the free energy for such models in a domain uniform in the size n of the matrix. It is the crux of the theorem uniform in size n. It relies on a new and simpler method which can be also applied in the general case of non-Hermitian matrix which was earlier treated by the same authors. Please, I urge you not to be shy, not to be afraid of interrupting me. Let the mu of H be Gaussian unitary ensemble independent and identically distributed measure with a covariance of 1 over n with normalization and a Gaussian behavior noted in the usual convention that you can see in the screen. The HP2P model is defined by an action defined by S of lambda in H equal lambda trace of H to the 2P, P greater than 3. The seminar theorem in the case P equal 2 was treated much earlier. It constitutes the birth of the loop vertex expansion. Now the partition function and a free energy of the model. The Z of lambda and n equals sum over d mu of exponential minus n S of lambda and n and f of lambda n equals 1 over n square log of n along z. Right k equals h square root of 1 plus lambda h to the 2P minus 2. So, k square equals h square plus lambda h to the 2P and put t equals h square of k square. The first catalan equation is z times tz to the p minus tz plus 1 equals 0. With z equals minus lambda k to the 2P minus 2. The sum of variable is z to k h of k equal k square root of t of z. Let us define f of lambda u equal square root of t of minus lambda u to the 2P minus 2 h of lambda of u equals u times the f lambda of u and k lambda of v equals v times square root of 1 plus lambda v to the 2P minus 2 so that the function h and k are inverse of each over jacobian a change of variable produces a new non-polynomial interaction dh over dk equals determinant of h times 1 minus 1 times h over k times 1 minus 1 times h over k. This is a formula with is a true quote generally. It can be exact through perturbation theory. It holds non-parturbative as well as possible. Then comes the main theorem of the paper. The theorem for epsilon greater than 0 there exists a small enough so that the expansion is absolutely the concessions and defines an analytic function of lambda uniformly bounded in the uniform in n parkman domain defined by p of epsilon and eta equals the set of zero like one modulus of lambda less than eta modulus of argument of lambda less than pi minus epsilon. The main interest in expansion explicitly are and absolutely convergent in the line of the loop vertex expansion. Now in the figure is the parkman domain with radius less than eta and angular size less than 2 pi minus 2 epsilon. We first we expand the partition function as an integral sum of an integral of the vertices as in perturbative theory. Then we apply the BKAR formula. The BKAR formula is a Taylor expansion formula with integral reminder in cerebral variable. The result is a sum of the set of a forest f on n labelled vertices of an integral. The main difficulty of this at first sight indeed daunting formula is that this integral is computed over a parameter x which 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 formula is a complex graph for example here is a picture of the first complex graphs up to order seven. Now for in example in the case n equal to there is two forests including the empty one and the BKAR formula is simply f of one equal f of zero plus sum of from zero up to one the two w times f prime of w. We recognize the formula at order one with an integral reminder. Now take another example for an equal free there are seven forests the it is a verb that the main formula appears for the first time. The formula is decomposed into the empty forest free singleton forest one one zero zero zero one zero and zero zero one we have a single parameter and two three doubletons forest one one zero one zero one zero one one with two parameters. One here is the least. A one doubletons forest in example is double sum from zero up to one one of w one w two times d square one two of f of two w one w two mean of w one and w two and so on. Introducing the uh contest notation la mauve that you are familiar with the BKAR formula and introducing the contest notation d mu d k forest and s n equal to the force of the a going to one to n f s of lambda and k k a we obtain some of the amplitude the amplitude a forest. We aware a forest is a part of a trivial factors an integral of a Gaussian of the derivative computed 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 sum of the same amplitude but made of spanning trees uh uh frost uh t uh n the statement if the partition function is made of some discrete object the the logarithm the logarithm is made of the same object but restricted to the connected case is a true for a much wider class and the class of graphs of the class of forest. It pertains to the class of componentoric species defined by André Joigal and developed by Canadian mathematician François Bergeron, Gilles Labelle and Pierre Le Roux. In essence it is an abstract systematic method for continued district structure for example a method of graphs of permutations of matroids uh uh uh to a lens of Carl Parger uh for uh of phi for models, any swimmer function S is expressed in a constructivier of the simply a rearrangement of the participative series in the endermic expansion in terms of square root of lambda phi square sigma S is given by a sum of G in G of W of G t H of G t equal when rearranged as a sum of T of T equal sum of G superset of T of W of G t H of G it's a simple rearrangement with the sum of the modulus of A t being now convergent when the three weights are defined by the percentage of the x-sector or one of which the cross-cultury is leading if you don't know what is a cross-cultury please ask in the set of question times at this moment the constructive theory of tensor models versus matrix model relies entirely on group vertex expansion there are a few list of works by a lot of authors a paper by Louis Caigoni and myself is especially good for noticing defined an analytic theorem for tensor with positive single trace of higher order in the figure two graphs are positive the first one does not symmetry axis of who is a melodic however for the moment if we do not know if a band is uniform along the expected domain optimal in N we do not even know what is what domain now I give you some ideas of the proof of the theorem we fix the tree with at least N greater than two nodes the case N equals one requires a spatial treatment the root g of u equals h of u minus u notice that g band is at lambda equals zero so that g of u equals the Taylor reminder of its derivative now we use factorization through allomorphic calculus f of h equals a circular integral of gamma of dv times f of ev over v minus h provided the contour gamma includes the full spectrum of h here are shown q-hole contour gamma and circling the spectrum of h which for its mission lies on the real axis here is a picture of the vertex with some of its corner operators we read all of k uk plus one there is only one operator k per vertex this is due to the fact that the formula b k a r depends only on the vertex therefore there is only one vector base and tensor a a tensor e b if the two borders factorize independently the operator o of k ck u k u k plus one is diagonal on the basis e a tensor e b with values o equals one other one plus sigma e b times three terms of the type one of one u minus mu plus three terms with the two tensor symmetric of each other now calling nu a equals h of mu a sigma is defined by one plus sigma to the power minus one equal k of mu a minus k of mu b over mu a minus mu b the label k executes the corresponding contour variable the upper left corner between this half eight indicated by the scope symbol contains a free one of u minus k operators with index indices k k and k plus one the following lemmas are a bit technical and i could give some explanation during the question time the theorem depends on who lemma is lemma one contour gamma will have the bound absolute value of g lambda of a u less than constant times lambda raised to the power one over four p squared times you can see in the screen the next lemmas are bound lemma two, lemma three, lemma five four and lemma five this is with the first preceding lemmas the proof is now complete it is worth noticing that the LVI as precise the prying energy constructively it has the great advantage of being a convergent sum when particular techniques in Feynman-Grass fail the trick of involving the parameter for each corner of each vertices is the key of the simplification made by this theorem it simplifies also the non Hermitian complex case or the case of symmetric or symplectic matrices as was said the case n equals one requires a special treatment it is actually quite subtle it involves five terms it's with a different structure i could give again some explanation during the question time who are your passions? thank you dear Vincent are there questions, remarks or comments about your proofs? thank you for giving a sketch of proofs so we have five minutes before the okay no remark, comment or is it is your talk preparing some tools for Joseph Ben-Gelun? Joseph is going to make some independent talks but the matrices are on ribbon graphs and Joseph made some interesting constructions having generalized ribbon graphs colored ribbon graphs there is a link for cancer graphs with the local Yoni talk work of myself not the matrix talk okay thank you dear Vincent I think we can have a little break until precisely Joseph's talk Joseph's talk is quite delayed by Sanjay talk in the journal club it is a talk called collide almost thank you for this pointer so I propose that we have a break until 15-20 okay and then we resume at precisely at 15-20 but of course as they are chat they chat you can chat together I have something to print in the meanwhile thank you very much