The $H^{2|2}$ monotonicity theorem revisited

THE H 2 ∣ 2 MONOTONICITY THEOREM REVISITED YICHA O HUANG AND XIAOLIN ZENG Abstract. W e use sup ersymmetric lo calization and integration b y parts to derive v aria- tional and con v ex correlation inequalities in statistical physics. As a primary application, w e give an alternativ e pro of of the monotonicity theorem for the H 2 ∣ 2 sup ersymmetric h yp erb olic sigma mo del. This recov ers a result of P oudevigne without relying on proba- bilistic couplings. 1. Introduction W e extend the metho d of sup ersymmetric lo calization to establish contin uous correla- tion inequalities in probabilit y and statistical physics. As a p edagogical warm-up, w e first revisit classical Gaussian comparison inequalities b efore applying this metho d to the H 2 ∣ 2 sup ersymmetric hyperb olic sigma mo del introduced b y Zirnbauer [ Zir91 ]. In particular, w e revisit the monotonicity theorem for the H 2 ∣ 2 mo del, originally established b y Poudevi- gne [ P A24 ] via discrete probabilistic couplings. Historically , lo calization formulas in supersymmetry hav e tak en v arious forms in b oth the physics and mathematics communities [ Wit88 , Wit92 , Nek03 , P es12 , DH82 , BV82 , AB84 ]. These formulas hav e broad applications in mathematical physics; in integrable mo dels, for example, sup ersymmetric reductions via lo calization form ulas yield exact solu- tions [ PZBea17 ]. Another imp ortan t class of applications for sup ersymmetric lo calization in volv es Morse-type dimension-comparison inequalities in geometry and top ology [ Wit82 ]. W e will introduce the precise sup ersymmetric definition of the H 2 ∣ 2 mo del later in the pap er: here w e state the H 2 ∣ 2 monotonicit y theorem as a prop ert y of real-v alued integrals, without using the sup ersymmetric language, following [ P A24 ]. Theorem 1 (Monotonicity theorem for the H 2 ∣ 2 mo del) . Fix an inte ger N ≥ 1 , c onsider the vertex set V = { 1 , . . . , N , δ } with a r o ot vertex δ , unoriente d e dges ( ij ) = ( j i ) for i, j ∈ V , and symmetric p ositive e dge weights W ij = W j i > 0 for i, j ∈ V . Define (1) D W ( t 1 , . . . , t N ) =  T ∈ T  ( ij ) ∈ T W ij e t i + t j K ey wor ds and phr ases. Sup ersymmetric Lo calization; Conv exity Inequalities; Horospherical Co ordi- nates; Sup ersymmetric In tegration by P arts. 1 2 YICHAO HUANG AND XIA OLIN ZENG for t = ( t 1 , . . . , t N ) and t δ = 0 , wher e the sum is over the sp anning tr e es T of the c omplete gr aph over { 1 , . . . , N , δ } . Consider the effe ctive H 2 ∣ 2 action functional S W eff ( t ) = 1 2  i,j ∈ V W ij ( cosh ( t i − t j ) − 1 ) − 1 2  ln D W ( t 1 , . . . , t N ) − 2 N  i = 1 t i  . Then for fixe d p 1 , . . . , p N > 0 , the inte gr al K =  R N f  N  i = 1 p i e t i  e − S W eff ( t ) dt 1 ⋯ dt N is non-incr e asing in e ach of the ( W ij ) i,j ∈ { 1 ,...,N ,δ } if f ∶ R → R is c onvex. An analogue of this monotonicit y theorem is conjectured for the H 2 ∣ 4 mo del [ BH23 , Equa- tion (5.5)]. T o establish monotonicity for the H 2 ∣ 2 mo del, Poudevigne [ P A24 ] developed a probabilistic metho d. Ho wev er, this metho d relies heavily on the sp ecific structure of the H 2 ∣ 2 setting, making a direct generalization to the H 2 ∣ 4 mo del difficult. Sp ecifically , three structural properties limit the broader application of this probabilistic approach. First, the pro of uses a graph reduction prop erty , which implies that studying a tw o-p oint graph suf- fices for the H 2 ∣ 2 mo del (see Remark 5 for a short alternativ e pro of assuming this property). Second, it relies on the triviality of the H 2 ∣ 2 partition function, whic h follo ws directly from the sup ersymmetric lo calization formula [ DSZ10 , Prop osition 2]. Third, it utilizes a dis- crete probabilistic coupling lemma that requires the first tw o prop erties, thereb y confining the technique to the H 2 ∣ 2 setting. This pap er pro vides an alternativ e, more general approach to the monotonicity theorem. P oudevigne’s approac h is motiv ated b y the exact probabilistic representation of the H 2 ∣ 2 mo del, building on the surprising discov ery of [ ST15 ] in connection to the V ertex Rein- forced Jump Pro cess (VRJP) and Edge Reinforced Random W alk (ERR W), for whic h he deriv ed the uniqueness of phase transition using the H 2 ∣ 2 monotonicit y theorem. Our study is based on the original sup ersymmetric represen tation of the H 2 ∣ 2 mo del: a contin uous sup ersymmetric v ariational calculation entirely replaces the discrete probabilistic coupling lemma, and our pro of av oids any sp ecific H 2 ∣ 2 graph reduction prop ert y . R emark 2 . With our new metho d, we slightly generalize Theorem 1 by relaxing the con- v exity condition on f , see Theorem 6 b elow. In particular, it is not necessary to assume the p ositivity p 1 , . . . , p N > 0 of the linear combination. 1.1. Ov erview of our metho d. The core of our metho d is the sup ersymmetric lo caliza- tion formula. Readers already familiar with this language can jump straight to Section 2.4 for a short, self-contained pro of of a slightly more general form of Theorem 1 . In mathematical physics, integrabilit y in sup ersymmetric theories is largely driven b y the lo calization form ula. Man y calculations in a sup ersymmetric theory can b e exactly reduced to lo cal con tributions or semi-classical approximations by lo calizing onto a low er- dimensional critical manifold. How ever, most applications of the sup ersymmetric lo cal- ization form ula concern exact equalities, and relatively few (to our kno wledge, none sys- tematically) deal with inequalities, esp ecially those inv olving contin uous parameters. Our THE H 2 ∣ 2 MONOTONICITY THEOREM REVISITED 3 starting p oint is the striking similarit y b et ween Theorem 1 and Kahane’s conv exity in- equalit y (see [ Hua25 , Section 3] for a sligh tly more general form) or Slepian’s inequality for Gaussian processes. The latter classical comparison inequalities can b e pro ved by showing a monotonicity prop erty under a rotation of the Euclidean co ordinate system. Since the sup ersymmetric generator Q squares to a rotation generator by Cartan’s magic form ula, it is natural to adapt the Gaussian interpolation metho d and to use Q to p erform (sup er- )in tegration b y parts to establish conv exit y or comparison inequalities in sup ersymmetric statistical physics mo dels. Although a generic sup ersymmetric integral lacks a deterministic sign, utilizing horospherical co ordinates resolves this issue. Switching b etw een coordinates and p erforming sup ersymmetric integration by parts in the correct setting results in a deterministic sign for the deriv ative even after integrating o ver the Grassmann v ariables. Although many calculations in this pap er are written using the language of sup ersym- metric in tegration, one could in principle translate them in to probabilistic real-v ariable calculations by writing out explicitly all the Berezin determinan ts. How ev er, we strongly feel that the sup ersymmetric setting is where these calculations live naturally . It is also in triguing to ask whether the sup ersymmetric metho d can fit naturally in the theory of matroids and Loren tzian p olynomials [ BH20 , Huh23 ]. A ckno wledgements. Y.H. is partially supp orted by the National Key R&D Program of China No. 2022YF A1006300 and NSFC-12301164. X.Z. ackno wledges the supp ort of the Institut de rec herche en mathématiques, interactions & applications: IRMIA++. 2. A shor t proof of the H 2 ∣ 2 monotonicity theorem 2.1. Sign conv en tion. Since we deal with inequalities in this pap er, w e must first establish our sign con ven tions for the Berezin sup ersymmetric calculus. W e denote b osonic v ariables b y Latin letters x, y , z , t, s, . . . , and fermionic (or Grassmann) v ariables by Greek letters ξ , η , ψ , ¯ ψ , . . . . W e indicate vectors with b oldface letters, such as z or ξ . Recall that Grassmann v ariables anti-comm ute, meaning ξ η = − η ξ . When a function F takes fermionic arguments, w e expand it using a T a ylor series. Be- cause Grassmann v ariables anti-comm ute, they are nilp otent (e.g., ξ 2 = 0 ). Consequen tly , an y T a ylor series in these v ariables truncates, meaning F is simply a p olynomial in the Grassmann v ariables. The left fermionic deriv ative acts as an o dd deriv ation. If F do es not dep end on ξ , the deriv ative satisfies ∂ ∂ ξ ( ξ F ) = F , ∂ ∂ ξ F = 0 . Analogous rules apply to all other Grassmann v ariables. The fermionic in tegral formally matc hes the fermionic deriv ativ e. F or any differen tiable function F , we ha v e  ∂ ξ F ( ξ ) = ∂ ∂ ξ F ( ξ ) . 4 YICHAO HUANG AND XIA OLIN ZENG An imp ortant consequence is the fermionic Gaussian integral: for any N × N matrix Σ , (2)  N  i = 1 ∂ ξ i ∂ η i e − ⟨ ξ , Σ η ⟩ = det ( Σ ) , where ξ = ( ξ 1 , . . . , ξ N ) and similarly for η . A distinguished sup ersymmetric op erator Q plays a crucial role in the sequel: (3) Q = ξ ∂ ∂ x + η ∂ ∂ y + x ∂ ∂ η − y ∂ ∂ ξ . This op erator in terchanges the b osonic and fermionic v ariables. It is easily seen that it annihilates the distinguished even v ariable (4) H = x 2 + y 2 + 2 ξ η in the sense that Q ( H ) = 0 , and that H is also Q -exact since (5) H = Q ( λ ) , λ = xη − y ξ . Let f and g b e homogeneous elemen ts of the Grassmann superalgebra. The Grassmann parit y  f  ∈ { 0 , 1 } indicates whether f is b osonic (  f  = 0 , an even element) or fermionic (  f  = 1 , an o dd element). The op erator Q satisfies the (sup er-)Leibniz rule Q ( f g ) = ( Qf ) g + ( − 1 ) ∣ f ∣ f Q ( g ) . W e also use the sup ersymmetric lo calization formula [ DSZ10 , Lemma 15]. F or any b ounded smo oth function f = f ( x, y , ξ , η ) , we ha v e (6)  dµ Q ( f ) = 0 , where the Q -inv arian t (flat) Berezin form is dµ = 1 2 π dx dy ∂ ξ ∂ η . This in v ariance directly yields a Q -integration b y parts formula. If f and g are smo oth b ounded functions of ( x, y , ξ , η ) , then (7)  dµ Q ( f ) g =  dµ Q ( f g ) − ( − 1 ) ∣ f ∣  dµ f Q ( g ) = − ( − 1 ) ∣ f ∣  dµ f Q ( g ) . The resulting sign dep ends strictly on the Grassmann parity  f  . 2.2. Slepian-t yp e inequalit y with sup ersymmetry. T o demonstrate the metho d in a familiar setting, we first prov e the standard conv exity inequality for a one-dimensional Gaussian v ariable. Let N b e a cen tered normal random v ariable with v ariance w − 1 > 0 . Prop osition 3 (The simplest Gaussian con vex inequality) . The inte gr al I =  R f ( x ) e − w 2 x 2 √ w √ 2 π dx = E [ f ( N )] is non-incr e asing in w > 0 if f ∶ R → R is c onvex. Jensen’s inequality yields a one-line pro of, but the goal here is to illustrate the imple- men tation of the sup ersymmetric lo calization formula ( 6 ). THE H 2 ∣ 2 MONOTONICITY THEOREM REVISITED 5 Pr o of. Recall that dµ = 1 2 π dxdy ∂ ξ ∂ η and the even v ariable H defined in ( 4 ), we claim that I =  dµf ( x ) e − w 2 ( x 2 + y 2 + 2 ξ η ) =  dµf ( x ) e − w 2 H . T o see this equality , integrating out the Grassmann v ariables using ( 2 ) yields a factor of w . In tegrating out the real v ariable y yields ∫ e − w 2 y 2 dy =  2 π  w . Com bining these with the flat measure 1 2 π dxdy , the effective measure for x is precisely w 2 π  2 π w dx =  w 2 π dx . T aking the deriv ative, w e get ∂ ∂ w I = − 1 2  dµf ( x ) H e − w 2 H = − 1 2  dµf ( x ) Q ( λe − w 2 H ) using that Q ( λ ) = H with λ = xη − y ξ as in ( 5 ) and Q ( H ) = 0 . Integrating by parts with resp ect to Q using ( 7 ), we get 1 2  dµQ ( f ( x )) λe − w 2 H = 1 2  dµf ′ ( x ) ξ λe − w 2 H = 1 2  dµf ′ ( x )( xξ η ) e − w 2 H . Define the fermionic v ariable ν = − xy ξ − y 2 η such that Q ( ν ) = ξ λ = xξ η . Then Q -integrating b y parts again yields 1 2  dµf ′ ( x ) Q ( ν ) e − w 2 H = − 1 2  dµQ ( f ′ ( x )) ν e − w 2 H = − 1 2  dµf ′′ ( x ) ξ ν e − w 2 H . But ξ ν = − y 2 ξ η , and integrating out the Grassmann v ariables ξ , η yields ∂ ∂ w I = − 1 2  R 2 dxdy 2 π f ′′ ( x ) y 2 e − w 2 ( x 2 + y 2 ) < 0 , since the last expression is an integration with real v ariables, f is conv ex, and y 2 > 0 . □ One chec ks after integrating out y that the deriv ativ e is exactly − 1 2 w 2 E [ f ′′ ( N )] , whic h is exp ected via the classical Stein’s lemma. 1 2.3. Sup ersymmetric pro of of the H 2 ∣ 2 monotonicit y theorem. As a further illus- tration, w e give a short sup ersymmetric pro of o f the monotonicity theorem for the H 2 ∣ 2 mo del. The original pro of of this result is due to Poudevigne [ P A24 , Theorem 6], who w orked directly with real-v alued integrals and relied on probabilistic to ols such as discrete coupling techniques and Jensen’s inequalit y . Our alternativ e pro of in this section concisely replaces the tec hnical coupling construction of P oudevigne [ P A24 , Section 3]. 1 The statemen t and proof ab ov e are easily generalized to any higher-dimensional Gaussian v ectors: one obtains in this manner the so-called dual-type Slepian Gaussian comparison inequalities (dual in the sense that we v ary the inv erse of the cov ariance matrix). W e omit the general computations. 6 YICHAO HUANG AND XIA OLIN ZENG 2.3.1. Definition and statement. W e recall the definition of the H 2 ∣ 2 sup ersymmetric hyper- b olic sigma mo del introduced by Zirnbauer [ Zir91 ] following the presentations of [ DSZ10 , ST15 , SZ19 , BCHS21 ]. W e define the sp ecial b osonic v ariable z i =  1 + x 2 i + y 2 i + 2 ξ i η i (defined via its terminating T a ylor series) to form the R 3 ∣ 2 spin v i = ( x i , y i , z i , ξ i , η i ) for i ∈ V = { 1 , . . . , N , δ } , with inner pro duct v i ⋅ v j = x i x j + y i y j − z i z j + ξ i η j + ξ j η i , so that the hyperb olic constraint v i ⋅ v i = − 1 is satisfied for all i ∈ V . W e also imp ose the b oundary condition v δ = ( 0 , 0 , 1 , 0 , 0 ) , the latter b eing the base p oint of the H 2 ∣ 2 manifold. The distinguished sup ersymmetric op erator Q is defined b y Q = N  i = 1 Q i = N  i = 1  ξ i ∂ ∂ x i + η i ∂ ∂ y i + x i ∂ ∂ η i − y i ∂ ∂ ξ i  . Giv en symmetric p ositive edge weigh ts W ij = W j i > 0 for i, j ∈ V , the H 2 ∣ 2 action is (8) S W = 1 4  i,j ∈ { 1 ,...,N ,δ } W ij ( v i − v j ) 2 = − 1 2  i,j ∈ { 1 ,...,N } W ij ( v i ⋅ v j + 1 ) +  i ∈ { 1 ,...,N } W iδ ( z i − 1 ) . As Q annihilates all inner pro ducts v i ⋅ v j and z i , this action is Q -closed. Finally , the (h yp erb olic) Q -inv arian t Berezin volume form for the H 2 ∣ 2 mo del is D µ =  N  l = 1 1 z l dµ l  =  N  l = 1 1 2 π z l dx l dy l ∂ ξ l ∂ η l  . The monotonicity theorem for the H 2 ∣ 2 mo del with the formulation of Theorem 1 has the following equiv alent description in the ab ov e Euclidean ( x, y , z , ξ , η ) co ordinates. Theorem 4 (Monotonicity theorem for H 2 ∣ 2 [ P A24 ]) . L et p 1 , . . . , p N ≥ 0 b e fixe d non- ne gative p ar ameters. F or any c onvex function f ∶ R → R , the fol lowing inte gr al with the action define d in ( 8 ) , J =  D µf  N  k = 1 p k ( x k + z k ) e − S W is non-incr e asing in the p ar ameters W ij for any i, j ∈ { 1 , . . . , N , δ } . T o make this statement rigorously well-defined, we recall the (sup er-)horospherical co- ordinates [ DSZ10 , Section 2.2], defined by the change of v ariables x i = sinh ( t i ) −  1 2 s 2 i + ¯ ψ i ψ i  e t i , y i = s i e t i , z i = cosh ( t i ) +  1 2 s 2 i + ¯ ψ i ψ i  e t i , ξ i = ¯ ψ i e t i , η i = ψ i e t i . (9) Under this change of v ariables, the hyperb olic measure D µ b ecomes the flat measure dµ ( t, s, ¯ ψ , ψ ) w eighted by ∏ N ℓ = 1 e − t ℓ . F or any smo oth b ounded function F ( v ) = F ( v 1 , . . . , v N ) , w e hav e the equalit y  D µF ( v ) =  dµ ( t, s, ¯ ψ , ψ )  N  l = 1 e − t l  F ( v ) , THE H 2 ∣ 2 MONOTONICITY THEOREM REVISITED 7 where dµ ( t, s, ¯ ψ , ψ ) is the flat Berezin form in the new co ordinates. In these co ordinates, conditional on the t -comp onents, the action functional decouples and b ecomes quadratic in the v ariables s and in ¯ ψ , ψ [ DSZ10 , Section 2.2]: S W = 1 2  i,j ∈ { 1 ,...,N } W ij  cosh ( t i − t j ) − 1 +  1 2 ( s i − s j ) 2 + ( ¯ ψ i − ¯ ψ j )( ψ i − ψ j ) e t i + t j  +  i ∈ { 1 ,...,N } W iδ  cosh t i − 1 +  1 2 s 2 i + ¯ ψ i ψ i  e t i  . (10) The corresp onding co v ariance matrix is the in verse of the N × N graph Laplacian ( ∆ W t ) , whic h is an M -matrix defined by (11) ( ∆ W t ) ij =        − W ij e t i + t j if i ≠ j, W iδ e t i + ∑ j ≠ i W ij e t i + t j if i = j. Therefore, Gaussian integration ov er the b osonic v ariables s pro duces a factor of ( det ∆ W t ) − 1 / 2 , while the fermionic Gaussian integration ov er ψ , ¯ ψ pro duces a factor of det ∆ W t . The net result is  det ∆ W t , whic h, by the Matrix-T ree Theorem, pro duces the  D W ( t ) term. T o- gether with the purely b osonic cosh terms from S W and the measure factor ∏ N ℓ = 1 e − t ℓ , this exactly reco v ers the real-v ariable effectiv e action S W eff ( t ) . Th us, integrating out all v ariables except t lea ves the real-v alued in tegral (12) J =  R N f  N  k = 1 p k e t k  dν δ ( t ) , where dν δ ( t ) = dν δ ( t 1 , . . . , t N ) is a probability measure on R N , called the effective t -field of H 2 ∣ 2 -mo del pinned at δ . In [ STZ17 , Equation (5)], the effectiv e t -field pinned at i 0 ∈ V = { 1 , 2 , . . . , N , δ } is defined as (13) dν i 0 ( t ) = 1 { t i 0 = 0 } e − 1 2 ∑ i,j ∈ V W ij ( cosh ( t i − t j ) − 1 )  D W ( t )  i ∈ V ,i ≠ i 0 e − t i dt i √ 2 π . This pro vides a setting in whic h one can apply a function f ∶ R → R to the sum ∑ k p k ( x k + z k ) = ∑ k p k e t k > 0 without sup ersymmetry , since the latter acquires a probabilistic meaning in the horospherical representation. 2.3.2. Example: r o ote d one-p oint gr aph. Consider the ro oted one-p oint graph with V = { 1 , δ } . W e assume p 1 = 1 by scaling and write, with W = W 1 δ and z = z 1 , the v ariation d dW  D µf ( x + z ) e − W ( z − 1 ) =  D µf ( x + z )( 1 − z ) e − W ( z − 1 ) = −  D µf ( x + z ) Q ( ˜ λ ) e − W ( z − 1 ) where ˜ λ = λ 1 + z with λ = xη − y ξ , so that Q ( ˜ λ ) = z − 1 . Integrating b y parts in Q , −  D µf ( x + z ) Q ( ˜ λ ) e − W ( z − 1 ) =  D µQ ( f ( x + z )) ˜ λe − W ( z − 1 ) =  D µf ′ ( x + z )( ξ ˜ λ ) e − W ( z − 1 ) . 8 YICHAO HUANG AND XIA OLIN ZENG Notice that ξ λ = Q ( ν ) where ν = − xy ξ − y 2 η . Therefore with ˜ ν = ν 1 + z w e get  D µf ′ ( x + z ) Q ( ˜ ν ) e − W ( z − 1 ) = −  D µf ′′ ( x + z ) ξ ˜ ν e − W ( z − 1 ) =  D µf ′′ ( x + z ) y 2 ξ η 1 + z e − W ( z − 1 ) . No w w e c hange to the horospherical co ordinates ( 9 ). Because the integrand already con- tains the top-degree Grassmann element ξ η , any additional fermionic terms will yield a pro duct containing ξ 2 or η 2 , whic h v anishes. Therefore, we can ev aluate z strictly at its ev en, scalar ‘b o dy’: bd ( z ) = cosh ( t ) + 1 2 s 2 e t > 0 . In tegrating out the fermions then yields  dtds 2 π ∂ ¯ ψ ∂ ψ e − t f ′′ ( e t ) s 2 e 4 t ¯ ψ ψ e − W ( z − 1 ) 1 + z = −  dtds 2 π e − t f ′′ ( e t ) s 2 e 4 t 1 + b d ( z ) e − W ( b d ( z ) − 1 ) ≤ 0 b y the conv exity of f and s 2 ≥ 0 . Hence, we ha ve pro ved that (14) d dW  D µf ( x + z ) e − W ( z − 1 ) ≤ 0 . This prov es the H 2 ∣ 2 monotonicit y theorem with N = 1 . R emark 5 . This inequality is a direct replacemen t for the discrete coupling argumen t of [ P A24 , Theorem 5]. Combined with a kno wn graph reduction prop erty for the H 2 ∣ 2 mo del [ P A24 , Prop osition 1.2.1] (see also [ SZ19 , Lemma 5] and [ L W20 ]), this already yields an alternative pro of of the H 2 ∣ 2 monotonicit y theorem. See App endix A for more details. 2.4. General H 2 ∣ 2 monotonicit y theorem. In the sequel, we use the new supersymmet- ric integration b y parts metho d to derive a simple and self-contained pro of of a slightly generalized H 2 ∣ 2 monotonicit y theorem, with a Kahane–Slepian type joint conv exity con- dition. This section is self-contained and do es not rely on any input from [ P A24 ]. Theorem 6 (Generalized form of the H 2 ∣ 2 monotonicit y theorem) . Supp ose F ∶ R N → R is smo oth and jointly c onvex (i.e., it has a p ositive semi-definite Hessian). Then the inte gr al  F ( e t 1 , . . . , e t N ) dν δ ( t ) is non-incr e asing in e ach of the e dge weights W ij with i, j ∈ { 1 , . . . , N , δ } . The smo othness assumption can b e dropp ed by standard approximation arguments. A sligh tly generalized form ulation of Theorem 1 follo ws at once since for any r e al p ar ameters p 1 , . . . , p N ∈ R without any sign condition, the function F ( e t 1 , . . . , e t N ) = f  N  k = 1 p k e t k  is jointly con vex in the v ariables e t 1 , . . . , e t N when f is conv ex. THE H 2 ∣ 2 MONOTONICITY THEOREM REVISITED 9 2.5. A useful lemma. W e now state a useful “switc hing lemma” to simplify many of the calculations in the sequel. In what follo ws, w e denote by x + z the v ector ( x 1 + z 1 , . . . , x N + z N ) , and b y ∂ k F and ∂ kℓ F the partial deriv atives of F . Lemma 7 (Switching lemma) . Consider a smo oth b ounde d function F ∶ R N → R and a Q -invariant action function A . F or any i, j, k ∈ { 1 , . . . , N } , we have  dµ F ( x + z ) x k ξ i η j A = −  dµF ( x + z ) x k y i y j A. A further Q -inte gr ation by p arts shows that this quantity also e quals N  ℓ = 1  dµ ∂ ℓ F ( x + z ) ξ ℓ η k y i y j A. Pr o of. Recall that the integral with resp ect to dµ of a fermionic form θ having unequal n umbers of ξ and η comp onents v anishes: ∫ dµ θ = 0 . W rite η j = Q ( y j ) , and apply the Q -integration by parts formula with the o dd v ariable F ( x + z ) x k ξ i and the Q -in v ariant function A :  dµ F ( x + z ) x k ξ i Q ( y j ) A =  dµ Q ( F ( x + z ) x k ξ i ) y j A. Applying the Leibniz rule yields Q ( F ( x + z ) x k ξ i ) = Q ( F ( x + z ) x k ) ξ i + F ( x + z ) x k Q ( ξ i ) . The first term v anishes up on in tegration b ecause it contains t wo more ξ comp onents than η comp onen ts. Using Q ( ξ i ) = − y i , we obtain −  dµ F ( x + z ) x k y i y j A, whic h prov es the first identit y . The second iden tity follows similarly . W rite x k = Q ( η k ) and p erform Q -integration b y parts with the Q -inv arian t function A : −  dµ F ( x + z ) Q ( η k ) y i y j A = −  dµ η k Q ( F ( x + z ) y i y j ) A. Expand the argument using the Leibniz rule: η k Q ( F ( x + z ) y i y j ) = η k Q ( F ( x + z )) y i y j + η k F ( x + z ) Q ( y i y j ) . The second term v anishes up on in tegration b ecause it carries an excess of tw o η comp onents. W e conclude that −  dµ η k Q ( F ( x + z ) y i y j ) A = N  ℓ = 1  dµ ∂ ℓ F ( x + z ) ξ ℓ η k y i y j A. This completes the pro of. □ W e no w prov e Theorem 6 . W e start by studying the v ariation with respect to a b oundary edge weigh t, then the v ariation in the bulk. 10 YICHAO HUANG AND XIA OLIN ZENG 2.5.1. Boundary variations. W e first pro ve the monotonicity with resp ect to a fixed b ound- ary edge w eight, say W 1 δ . Pr o of of The or em 6 in the b oundary c ase. Without loss of generality , consider the v aria- tion with resp ect to W 1 δ . With λ 1 = x 1 η 1 − y 1 ξ 1 as in the previous examples, ∂ ∂ W 1 δ J =  D µ F ( x + z )( 1 − z 1 ) e − S W =  D µ F ( x + z ) Q  − λ 1 1 + z 1  e − S W = N  k = 1  D µ ∂ k F ( x + z ) ξ k λ 1 1 + z 1 e − S W . A t this stage one can replace λ 1 b y x 1 η 1 , since the other part of λ 1 will create an excess of ξ comp onents and v anishes up on integration. Then we apply the switching lemma with the Q -inv arian t function A =  ∏ N ℓ = 1 1 z ℓ  1 1 + z 1 e − S W : ∂ ∂ W 1 δ J = N  k = 1  D µ ∂ k F ( x + z ) ξ k x 1 η 1 1 + z 1 e − S W = N  k,ℓ = 1  D µ ∂ kℓ F ( x + z ) ξ ℓ y k η 1 y 1 1 + z 1 e − S W . Using the expansion 1 1 + z 1 = ∫ ∞ 0 dαe − 2 α e − α ( z 1 − 1 ) , the ab o ve integral can b e rewritten as  ∞ 0 dαe − 2 α       N  k,ℓ = 1  D µ ∂ kℓ F ( x + z ) ξ ℓ y k η 1 y 1 e − S W,α       , where the action S W,α is defined with a shifted weigh t W 1 δ ↦ W 1 δ + α . It remains to show that for an y fixed α > 0 , the following quan tity is non-p ositive: N  k,ℓ = 1  D µ ∂ kℓ F ( x + z ) ξ ℓ y k η 1 y 1 e − S W,α . In the horospherical co ordinates ( 9 ) and dµ = ∏ N i = 1 dt i ds i ∂ ψ i ∂ ¯ ψ i 2 π , the ab o ve display becomes N  k,ℓ = 1  dµ e − ∑ i t i ∂ kℓ F ( e t ) ¯ ψ ℓ e t ℓ s k e t k ψ 1 e t 1 s 1 e t 1 e − S W,α . Gaussian integration in s, ¯ ψ , ψ using the cov ariance matrix G W,α t = ( ∆ W,α t ) − 1 recalled in ( 11 ) (with the α -shift in W 1 δ as ab ov e) yields, with the effective measure ( 12 ) but shifted by α , N  k,ℓ = 1  dν α δ ( t )  − e t ℓ + t 1 ( G W,α t ) ℓ 1 ∂ kℓ F ( e t ) e t k + t 1 ( G W,α t ) k 1  . The minus sign comes from the fermionic exp ectation E [ ¯ ψ i ψ j ] = − ( G W,α t ) ij . This is man- ifestly non-p ositive by the joint conv exity of F , since the term inside the square brack ets tak es the form − v T ( Hess ( F )) v . □ THE H 2 ∣ 2 MONOTONICITY THEOREM REVISITED 11 R emark 8 . Notice that the p ositivit y of the co efficien ts of the Green matrix G is not used. Therefore, our proof do es not rely on the M -matrix prop ert y of the H 2 ∣ 2 mo del, which was crucial for probabilistic metho ds such as random path expansion. 2.5.2. Bulk variations. W e now prov e the monotonicity with respect to the parameter W ij for a fixed pair i, j ∈ { 1 , . . . , N } : without loss of generality , suppose that ( i, j ) = ( 1 , 2 ) . The basic idea is to use a “rero oting” formula to shift the pinning condition from the vertex δ to the vertex 1 . After this rero oting, W 12 b ecomes a b oundary edge weigh t and we can directly inv ok e the previous pro of in the b oundary case. Lemma 9 (Rero oting formula) . L et δ, b ∈ V and c onsider the me asur es dν δ and dν b with differ ent pinning p oints δ and b , define d in ( 12 ) . If the variables t ′ and t ar e r elate d by t ′ i = t i − t b for al l i ∈ V , then e t b − t δ dν δ ( t ) = dν b ( t ′ ) . Pr o of. Recall from [ STZ17 , Equation (5)] that dν i 0 ( t ) = 1 { t i 0 = 0 } e − 1 2 ∑ i,j W ij ( cosh ( t i − t j ) − 1 )  D W ( t )  i ∈ V ,i ≠ i 0 e − t i dt i √ 2 π . The pro of follows by p erforming the change of v ariables t ′ i = t i − t b , and noticing that the scaling of  D W ( t ) defined in ( 1 ) exactly cancels that of the ∏ e − t i factor in the densit y . □ Recall that if F ( x 1 , . . . , x N ) is jointly con vex, its p ersp ective function P ( x 1 , . . . , x N , T ) = T ⋅ F ( x 1 T , . . . , x N T ) is join tly conv ex in ( x 1 , . . . , x N , T ) for T > 0 , see App endix B . Pr o of of The or em 6 in the bulk c ase. Consider the v ariation with resp ect to the bulk edge w eight W 12 of the quan tity J =  F ( e t ) dν δ ( t ) . W e rero ot the integral from δ to 1 : b y Lemma 9 we get J =  e t ′ δ F ( e t ′ ⋅ e − t ′ δ ) dν 1 ( t ′ ) , where the new pinning is t ′ 1 = 0 and the integration is o ver the v ariables t ′ i for i ∈ { 2 , . . . , N , δ } . Setting T = e t ′ δ , we see that the new function G ( e t ′ ) = e t ′ δ F ( e t ′ ⋅ e − t ′ δ ) is jointly con vex in the v ariables e t ′ 2 , . . . , e t ′ N , e t ′ δ . Since W 12 b ecomes a b oundary edge w eight after the rero oting, the generalized monotonicity theorem in Section 2.5.1 applies, and the pro of in the bulk case is complete. □ R emark 10 (T ow ards general H 2 n ∣ 2 m monotonicit y theorems) . One chec ks that most of the in termediate steps in the new sup ersymmetric pro of of the H 2 ∣ 2 monotonicit y theorem gen- eralize already to any H 2 n ∣ 2 m mo dels, especially the rero oting lemma which reduces a bulk v ariation to a b oundary v ariation. The only non-trivial p oint is a sup ersymmetric pro of of the b oundary v ariation for general H 2 n ∣ 2 m mo del: this is currently under inv estigation. 12 YICHAO HUANG AND XIA OLIN ZENG Appendix A. Replacing Poudevigne’s coupling argument by supersymmetric integra tion by p ar ts While Section 2.4 gives a self-contained pro of of the slightly more general H 2 ∣ 2 mono- tonicit y theorem, w e demonstrate here ho w the 1 -p oint inequality (Section 2.3.2 ) serv es as a direct drop-in replacement for P oudevigne’s discrete coupling lemma [ P A24 , Theorem 5]. This result cleanly slots into the existing graph-reduction framework [ STZ17 , SZ19 ]. Equation ( 14 ), written in the H 2 ∣ 2 horospherical co ordinates ( 9 ), is the statemen t that for any con vex function f ∶ R → R , (15) d dW  R f ( e t ) e − W ( cosh ( t ) − 1 ) √ W e t e − t dt √ 2 π ≤ 0 . W e in tro duce the random Schrödinger represen tation to leverage the graph reduction prop- ert y of the H 2 ∣ 2 mo del. Define the random Schrödinger op erator H β on a finite graph V with symmetric p ositiv e edge weigh ts W ij > 0 . Set H β ( i, i ) = 2 β i and H β ( i, j ) = − W ij for i ≠ j . [ STZ17 , Definition 1] in tro duced a probabilit y measure q W, 1 V ( dβ ) ; as the off-diagonal en tries of H β are deterministic, q W, 1 V ( dβ ) is also the la w of the random matrix H β . W e apply this setup to the t w o-vertex graph V = { 1 , δ } with edge weigh t W = W 1 δ . By [ STZ17 , Theorem 3], we can rewrite this horospherical integral ( 15 ) as an exp ectation o ver a random v ariable β = ( β 1 , β δ ) . Specifically , following [ STZ17 , Definition 1], let q W, 1 { 1 ,δ } ( dβ ) = 1 { H β > 0 } 2 π e W e − β 1 − β δ  det H β dβ 1 dβ δ . Let G = H − 1 β denote the Green function. F or an y constants p 1 , p δ ∈ R and an y conv ex function g ∶ R → R , define the conv ex function f by f ( y ) = g ( p δ + p 1 y ) . In [ STZ17 , The- orem 3], the distributional identit y e t = G ( 1 ,δ ) G ( δ,δ ) w as established, which yields an equiv alent reform ulation of ( 15 ): (16) d dW  g  p δ + p 1 G ( 1 , δ ) G ( δ, δ )  q W, 1 { 1 ,δ } ( dβ ) ≤ 0 . On a general graph with V = { 1 , 2 , . . . , N , δ } , let q W, 1 V ( dβ ) denote the distribution for the ( N + 1 ) × ( N + 1 ) random Schrödinger matrix H β . Let G = H − 1 β b e its inv erse. The explicit density of H β w as given in [ STZ17 , Definition 1], but w e only need the follo wing graph reduction prop ert y [ SZ19 , Lemma 5]: Prop osition 11. Partition the vertex set V into two subsets: the bulk V 1 = { 1 , . . . , N − 1 } and the b oundary V 2 = { N , δ } . 2 This p artition induc es a blo ck de c omp osition of the symmetric matric es H β and G : H β =   H V 1 ,V 1 β H V 1 ,V 2 β H V 2 ,V 1 β H V 2 ,V 2 β   , G =  G V 1 ,V 1 G V 1 ,V 2 G V 2 ,V 1 G V 2 ,V 2  . 2 The b oundary was { 1 , δ } in Section 2.5.1 ; we choose { N , δ } here to cleanly p erform the blo ck decomp osition. THE H 2 ∣ 2 MONOTONICITY THEOREM REVISITED 13 Then c onditional ly on ( β i ) i ∈ V 1 , ( G V 2 ,V 2 ) − 1 is distribute d ac c or ding to q  W N δ , 1 { N ,δ } , wher e  W N δ = W N δ +  H V 2 ,V 1 β ( H V 1 ,V 1 β ) − 1 H V 1 ,V 2 β  N δ . In p articular, d  W N δ dW N δ = 1 . Observ e that G ( δ, δ ) = G V 2 ,V 2 ( δ, δ ) and G ( N , δ ) = G V 2 ,V 2 ( N , δ ) . Let p 1 , . . . , p N , p δ b e real n umbers. [ STZ17 , Theorem 3] provides the identit y in law (17)  j ∈ V p j e t j = ∑ j ∈ V p j G ( j, δ ) G ( δ, δ ) . Decomp ose the n umerator sum o ver the subsets V 1 and V 2 :  j ∈ V p j G ( j, δ ) =  j ∈ V 1 p j G ( j, δ ) + p N G V 2 ,V 2 ( N , δ ) + p δ G V 2 ,V 2 ( δ, δ ) . In vert the blo ck matrix using the Sch ur complemen t form ula. This yields G V 1 ,V 2 = M G V 2 ,V 2 with M = − ( H V 1 ,V 1 β ) − 1 H V 1 ,V 2 β . The columns of M map to the b oundary vertices { N , δ } . Notice that G ( j, δ ) = [ M G V 2 ,V 2 ] j δ and we ha ve  j ∈ V 1 p j G ( j, δ ) =  j ∈ V 1 p j  M j N G V 2 ,V 2 ( N , δ ) + M j δ G V 2 ,V 2 ( δ, δ )  . Define α N = ∑ j ∈ V 1 p j M j N and α δ = ∑ j ∈ V 1 p j M j δ . Grouping the terms b y G V 2 ,V 2 comp onen ts,  j ∈ V p j G ( j, δ ) = ( α N + p N ) G V 2 ,V 2 ( N , δ ) + ( α δ + p δ ) G V 2 ,V 2 ( δ, δ ) . W e now define the shifted parameters ˜ p N = α N + p N and ˜ p δ = α δ + p δ . The original ratio ( 17 ) collapses into a 2 -p oint b oundary form: ∑ j ∈ V p j G ( j, δ ) G ( δ, δ ) = ˜ p δ + ˜ p N G V 2 ,V 2 ( N , δ ) G V 2 ,V 2 ( δ, δ ) . This algebraic identit y constitutes the core of [ P A24 , Lemma 2.2.2]. Moreo v er, α N , α δ dep end only on β 1 , . . . , β N − 1 , since M is entirely determined by H V 1 ,V 1 β and H V 1 ,V 2 β . Finally , Prop osition 11 yields that ( G V 2 ,V 2 ) − 1 follo ws the la w q  W N δ , 1 { N ,δ } asso ciated with the effectiv e edge weigh t  W N δ satisfying d  W N δ dW N δ = 1 . Apply the 2 -p oin t inequality ( 16 ), d dW N δ E q W, 1 V  g  ˜ p δ + ˜ p N G V 2 ,V 2 ( N , δ ) G V 2 ,V 2 ( δ, δ )   β 1 , . . . , β N − 1  ≤ 0 . T aking the exp ectation of this conditional inequality o ver the bulk v ariables β 1 , . . . , β N − 1 yields the monotonicit y theorem of the H 2 ∣ 2 mo del on the graph V = { 1 , . . . , N , δ } . 14 YICHAO HUANG AND XIA OLIN ZENG Appendix B. Perspective function of a jointl y convex function Consider a join tly conv ex function f ∶ R N → R . This is equiv alent to saying that, ∀ x , y ∈ R N , ∀ λ ∈ ( 0 , 1 ) , f ( λ x + ( 1 − λ ) y ) ≤ λf ( x ) + ( 1 − λ ) f ( y ) . Define its p ersp ectiv e function P ( x 1 , . . . , x N , T ) = T ⋅ F ( x 1 T , . . . , x N T ) on R N × R > 0 . 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Beijing Institute of Technology, School of Ma thema tics and St a tistics, Beijing, China Email address : yichao.huang@outlook.com Institut de Recherche Ma théma tique A v ancée, Université de Strasbourg, Strasbourg, France Email address : zeng@math.unistra.fr